ae L 2 Hodge Theory and Vanishing Theorems Jean-Pierre Demailly Universit'e de Grenoble I, Institute Fourier, BP 74, 38402 Saint-Martin d'H`res, France 1 j.-p. demailly, L 2 hodge theory and vanishing theorems 2 0. Introduction The aim of these notes is to describe two fundamental applications of L 2 Hilbert space techniques to analytic or algebraic geometry: Hodge theory, and the theory of L 2 estimates for the @ operator. The point of view adopted here is essentially analytic. The first part is focussed on Hodge theory and it is intended to be rather introductory. Thus the reader will find here only the most elementary topics, mostly those due to W.V.D. Hodge himself [Hod41] or to A. Weil [Wei57]. Hodge theory, as first conceived by its creator, consists of the study of the cohomology of Riemannian or K"ahlerian manifolds, by means of a description of harmonic forms and their properties. We refer to the treatment of J. Bertin-Ch. Peters [BePe95] and L. Illusie [Ill95] for a presentation of more advanced topics and applications (variation of Hodge structure, application of periods, Hodge theory in characteristic p ? 0 : : : ). We consider a Riemannian manifold X and a Euclidean or Hermitian bundle E over X. We assume that E is equipped with a connection D compatible with the metric: A connection is by definition a differential operator analogous to exterior differentiation, acting on forms of arbitrary degree with values in E, and satisfies Leibniz rule for the exterior product. The Laplace-Beltrami operator is the self-adjoint differential operator of second order \Delta E = DED \Lambda E + D \Lambda E DE, where D \Lambda E is the Hilbert space adjoint of DE. One easily shows that \Delta E is an elliptic operator. The finiteness theorem for elliptic operators shows then that the space H q (X; E) of harmonic q-forms with values in E is finite dimensional if X is compact (we say that a form u is harmonic if \Delta Eu = 0). If we assume in addition that the connection satisfies D 2 E = 0, the operator DE acting on forms of all degrees defines a complex called the de Rham complex with values in the local system of coefficients defined by E. The corresponding cohomology groups will be denoted by H q DR (X; E). The fundamental observation of Hodge theory is that any cohomology class contains a unique harmonic representative, since X is compact. It leads then to an isomorphism, called the Hodge isomorphism (0.1) H q DR (X; E) ' H q DR (X; E): When the manifold X and the bundle E are holomorphic, there exists a unique connection DE called the Chern connection, compatible with the Hermitian metric on E and has the following properties: DE splits into a sum DE = D 0 E + D 00 E of a connection D 0 E of type (1; 0) and a connection D 00 E of type (0; 1), such that D 02 E = D 002 E = 0 and D 0 E D 00 E + D 00 E D 0 E = \Theta (E) (Chern curvature tensor of the bundle). The operator D 00 E acting on the forms of bidegree (p; q) defines then for fixed p, a complex called the Dolbeault complex. When X is compact, the Dolbeault cohomology groups H p;q (X; E) satisfy a Hodge isomorphism analogous to (0:1), namely (0.2) H p;q (X; E) ' H p;q (X; E); where H p;q (X; E) denotes the space of harmonic (p; q)-forms with values in E, relative to the anti-holomorphic Laplacian \Delta 00 E = D 00 E D 00\Lambda E + D 00\Lambda E D 00 E . By utilizing this latter result, one easily proves the Serre duality theorem (0.3) H p;q (X; E) \Lambda ' H n\Gamma p;n\Gamma q (X; E \Lambda ); n = dimC X; 3 0. introduction which is the complex version of the Poincar'e duality theorem. The central theorem of Hodge theory concerns compact K"ahler manifolds: A Hermitian manifold (X; !) is called K"ahlerian if the Hermitian (1; 1)-form ! = i P j;k !jkdzj ^ dzk satisfies d! = 0. A fundamental example of a compact K"ahlerian manifold is given by the projective algebraic manifolds. If X is compact K"ahlerian and if E is a local system of coefficients on X, the Hodge decomposition theorem asserts that H k DR (X; E) = M p+q=k H p;q (X; E) (Hodge decomposition)(0.4) H p;q (X; E) ' H q;p (X; E \Lambda ): (Hodge symmetry)(0.5) The intrinsic character of the decomposition will be shown here in a somewhat original way, via the utilization of the Bott-Chern cohomology groups (@@-cohomology groups). It follows from these results that the Hodge numbers h p; q = dimC H p;q (X; C ) satisfy the symmetry property h p;q = h q;p = h n\Gamma p;n\Gamma q = h n\Gamma q;n\Gamma p , and that they are connected to the Betti numbers bk = dimC H k DR (X; C ) by the relation bk = P p+q=k h p;q . A certain number of other remarkable cohomological properties of compact K"ahler manifolds are obtained by means of the primitive decomposition and the hard Lefschetz theorems (which in turn is a result of the existence of an sl(2; C ) action on harmonic forms). These results allow us to describe in a precise way the structure of the Picard group Pic(X) = H 1 (X; O \Lambda ) in the K"ahlerian case. In a more general setting, we discuss the Hodge-Fr"olicher spectral sequence (the spectral sequence connecting Dolbeault to de Rham cohomology), and we show how one can utilize this spectral sequence to obtain some general results on the Hodge numbers h p;q of compact complex manifolds. Finally, we establish the semi-continuity of the dimension of the cohomology groups H q (Xt; Et) of bundles arising from a proper and smooth holomorphic fibration X ! S (result due to Kodaira-Spencer), and we deduce from it that the Hodge numbers h p;q (Xt) are constant if the fibers Xt are K"ahlerian (invariance of the h p;q under deformations); the holomorphic nature of the Hodge filtration F p H k (Xt; C ) = \Phi r*pH r;k\Gamma r (Xt; C ) relative to the Gauss-Manin connection is proven by means of the theorem on the coherence of direct images, applied to the relative de Rham complex \Omega ffl X=S of X ! S. In the second part, after recalling some of the relevant concepts of positivity and pseudoconvexity, we establish the Bochner-Kodaira-Nakano identity connecting the Laplacians \Delta 0 E and \Delta 00 E . The identity in question furnishes an explicit expression of the difference \Delta 00 E \Gamma \Delta 0 E in terms of the curvature \Theta (E) of the bundle. Under adequate hypothesis (weak pseudoconvexity of X, positivity of the curvature of E), one arrives a priori at the estimate jjD 00 E ujj 2 + jjD 00\Lambda E ujj 2 * Z X *(z)juj 2 dV (z) where * is a positive function depending on the eigenvalues of curvature. The inequality is valid here for any form u of bidegree (n; q); n = dim X; q * 1, with values in E, u belonging to the Hilbert space domains of D 00 E and D 00\Lambda E . By an argument of Hilbert space duality one deduces from this the following fundamental theorem, essentially due to H"ormander [H"or65] and Andreotti-Vesentini [AV65]: 0.6. Theorem. Let (X; !) be a K"ahler manifold, dim X = n. Assume that X is weakly pseudoconvex. Let E be a Hermitian line bundle and suppose that the j.-p. demailly, L 2 hodge theory and vanishing theorems 4 eigenvalues of the curvature form i\Theta (E) with respect to the metric ! at each point x 2 X, satisfy fl1(x) ^ \Delta \Delta \Delta ^ fln(x): Further, suppose that the curvature is semi-positive, i.e. fl1 * 0 everywhere. Then for any form g 2 L 2 (X; \Lambda n;q T \Lambda X \Omega E) such that D 00 E g = 0 and Z X (fl1 + \Delta \Delta \Delta + flq) \Gamma 1 jgj 2 dV! ! +1; there exists f 2 L 2 (X; \Lambda n;q\Gamma 1 T \Lambda X \Omega E) such that D 00 E f = g and Z X jf j 2 dV! ^ Z X (fl1 + \Delta \Delta \Delta + flq) \Gamma 1 jgj 2 dV!: An important observation is that the above theorem still remains valid when the metric h of E acquires singularities. The metric h is then given in each chart by a weight e \Gamma 2' associated to a plurisubharmonic function ' (by definition ' is psh if the matrix of second derivatives (@ 2 '=@zj@zk), calculated in the sense of distributions, is semi-positive at each point). Taking into account Theorem (0.6), it is natural to introduce the multiplier ideal sheaf J (h) = J ('), made up of the germs of holomorphic functions f 2 OX;x such that R V jf j 2 e \Gamma 2' converges in a sufficiently small neighbourhood V of x. A recent result of A. Nadel [Nad89] guarantees that J (') is always a coherent analytic sheaf, whatever the singularities of '. In this context, one deduces from (0.6) the following qualitative version, concerning the cohomology with values in the coherent sheaf O(KX \Omega E) \Omega J (h) (KX = \Lambda n T \Lambda X being the canonical bundle of X). 0.7. Nadel Vanishing Theorem ([Nad89], [Dem93b]). Let (X; !) be a weakly pseudoconvex K"ahler manifold, and let E be a holomorphic line bundle over X equipped with a singular Hermitian metric h of weight '. Suppose that there exists a continuous positive function ffl on X such that the curvature satisfies the inequality i\Theta h(E) * ffl! in the sense of currents. Then H q (X; O(KX \Omega E) \Omega J (h)) = 0 for all q * 1: In spite of the relative simplicity of the techniques involved, it is an extremely powerful theorem, which by itself contains many of the most fundamental results of analytic or algebraic geometry. Theorem (0.7) also contains the solution of the Levi problem (equivalence of holomorphic convexity and pseudoconvexity), the vanishing theorems of Kodaira-Serre, Kodaira-Akizuki-Nakano and Kawamata-Viehweg for projective algebraic manifolds, as well as the Kodaira embedding theorem characterizing these manifolds among the compact complex manifolds. By its intrinsic character, the "analytic" statement of Nadel's theorem appears useful even for purely algebraic applications. (The algebraic version of the theorem, known as the Kawamata-Viehweg vanishing theorem, utilizes the resolution of singularities and does not give such a clear description of the multiplier sheaf J (h).) In a recent work [Siu96], Y.T. Siu has shown the following remarkable result, by utilizing only the Riemann-Roch formula and an inductive Noetherian argument for the multiplier sheaves. The technique is described in x16 (with some improvements developed in [Dem96]). 5 0. introduction 0.8. Theorem [Siu96], [Dem96]). Let X be a projective manifold and L an ample line bundle (i.e. has positive curvature) on X. Then the bundle K \Omega 2 X \Omega L \Omega m is very ample for m * m0(n) = 2 + \Gamma 3n+1 n \Delta , where n = dim X. The importance of having an effective bound for the integer m0(n) is that one can also obtain embeddings of manifolds X in projective space, with a precise control of the degree of the embedding. As a consequence of this, one has a rather simple proof of a significant finiteness theorem, namely "Matsusaka's big theorem" (cf. [Mat72], [KoM83], [Siu93], [Dem96]): 0.9. Matsusaka's Big Theorem. Let X be a projective manifold and L an ample line bundle over X. There exists an explicit bound m1 = m1(n; L n ; KX\Delta L n\Gamma 1 ) depending only on the dimension n = dim X and on the first two coefficients of the Hilbert polynomial of L, such that mL is very ample for m * m1. From this theorem, one easily deduces numerous finiteness results, in particular the fact that there exist only a finite number of families of deformations of polarized projective manifolds (X; L), where L is an ample line bundle with given intersection numbers L n and KX \Delta L n\Gamma 1 . Part I: L 2 Hodge Theory 1. Vector bundles, connections and curvature The goal of this section is to recall some basic definitions of Hermitian differential geometry with regard to the concepts of connection, curvature and the first Chern class of line bundles. 1.A. Dolbeault cohomology and the cohomology of sheaves. Assume given X a C -analytic manifold of dimension n. We denote by \Lambda p;q T \Lambda X the bundle of differential forms of bidegree (p; q) on X, i.e. differential forms which can be written u = X jIj=p; jJj=q uI;J dz ^ zJ ; dzI := dzi 1 ^ \Delta \Delta \Delta ^ dzi p ; dzJ := dzj 1 ^ \Delta \Delta \Delta ^ dzj q ; where (z1; : : : ; zn) are local holomorphic coordinates, and where I = (i1; : : : ; ip) and J = (j1; : : : ; jq) are multi-indices (increasing sequences of integers in the interval [1; : : : ; n], with lengths jIj = p; jJj = q). Let A p;q be the sheaf of germs of differential forms of bidegree (p; q) with complex valued C 1 coefficients. We recall that the exterior derivative d decomposes into d = d 0 + d 00 where d 0 u = X jIj=p;jJj=q;1^k^n @uI;J @zk dzk ^ dzI ^ dzJ ; d 00 u = X jIj=p;jJj=q;1^k^n @uI;J @zk dzk ^ dzI ^ dzJ are of type (p+1; q), (p; q+1) respectively. The well known Dolbeault-Grothendieck Lemma asserts that all d 00 -closed forms of type (p; q) with q ? 0 are locally d 00 -exact (this is the analogue for d 00 of the usual Poincar'e Lemma for d, see for example [Hor66]). In other words, the complex of sheaves (A p;ffl ; d 00 ) is exact in degree q ? 0: and in degree q = 0, Ker d 00 is the sheaf \Omega p X of germs of holomorphic forms of degree p on X. More generally, if E is a holomorphic vector bundle of rank r over X, there exists a natural operator d 00 acting on the space C 1 (X; \Lambda p;q T \Lambda X \Omega E) of C 1 (p; q)- forms with values in E. Indeed, if s = P 1^*^r s*e* is a (p; q)-form expressed in terms of a local holomorphic frame of E, we can define d 00 s := P (d 00 s*)\Omega e*; by first observing that the transition matrices corresponding to a change of holomorphic frame are holomorphic, and which commute with the operation of d 00 . It then follows that the Dolbeault-Grothendieck Lemma still holds for forms with values in E. For every integer p = 0; 1; : : :; n, the Dolbeault cohomology groups H p;q (X; E) are defined as being the cohomology of the complex of global forms of type (p; q) (indexed by q): (1.1) H p;q (X; E) = H q (C 1 (X; \Lambda p;ffl T \Lambda X \Omega E)): There is the following fundamental result of sheaf theory (de Rham-Weil Isomorphism Theorem): Let (L ffl ; ffi) be a resolution of a sheaf F by acyclic sheaves, i.e. a complex (L ffl ; ffi) given by an exact sequence of sheaves 0 ! F j \Gamma ! L 0 ffi 0 \Gamma ! L 1 ! \Delta \Delta \Delta ! L q ffi q \Gamma ! L q+1 ! \Delta \Delta \Delta ; 6 7 1. vector bundles, connections and curvature where H s (X; L q ) = 0 for all q * 0 and s * 1. (To arrive at this latter condition of acyclicity, it is enough for example that the L q are flasque or soft, for example a sheaf of modules over the sheaf of rings C 1 .) Then there is a functorial isomorphism (1.2) H q (\Gamma (X; L ffl )) ! H q (X; F): We apply this in the following situation. Let A p;q (E) be the sheaf of germs of C 1 sections of \Lambda p;q T \Lambda X \Omega E. Then (A p;ffl (E); d 00 ) is a resolution of the locally free OXmodule \Omega p X \Omega O(E) (Dolbeault-Grothendieck Lemma), and the sheaves A p;q (E) are acyclic as C 1 -modules. According to (1.2), we obtain 1.3. Dolbeault Isomorphism Theorem (1953). For all holomorphic vector bundles E on X, there exists a canonical isomorphism H p;q (X; E) ' H p (X; \Omega p X \Omega O(E)): If X is projective algebraic and if E is an algebraic vector bundle, the theorem of Serre (GAGA) [Ser56] shows that the algebraic cohomology groups H q (X; \Omega p X \Omega O(E)) computed via the corresponding algebraic sheaf in the Zariski topology are isomorphic to the corresponding analytic cohomology groups. Since our point of view here is exclusively analytic, we will no longer need to refer to this comparison theorem. 1.B. Connections on differentiable manifolds. Assume given a real or complex C 1 vector bundle E of rank r on a differentiable manifold M of class C 1 . A connection D on E is a linear differential operator of order 1 D : C 1 (M; \Lambda q T \Lambda M \Omega E) ! C 1 (M; \Lambda q+1 T \Lambda M \Omega E) such that D satisfies Leibnitz rule: (1.4) D(f ^ u) = df ^ u + (\Gamma 1) deg f f ^ Du for all forms f 2 C 1 (M; \Lambda p T \Lambda M ); u 2 C 1 (X; \Lambda q T \Lambda M \Omega E). On an open set \Omega ae M where E admits a trivialization o/ : Ej \Omega ' \Gamma ! \Omega \Theta C r , a connection D can be written Du 'o/ du + \Gamma ^ u where \Gamma 2 C 1 (\Omega ; \Lambda 1 T \Lambda M \Omega Hom(C r ; C r )) is a given matrix of 1-forms and where d acts componentwise on u 'o/ (u*)1^*^r. It is then easy to verify that D 2 u 'o/ (d\Gamma + \Gamma ^ \Gamma ) ^ u on \Omega : Since D 2 is a globally defined operator, there exists a global 2-form (1.5) \Theta (D) 2 C 1 (M; \Lambda 2 T \Lambda M \Omega Hom(E; E)) such that D 2 u = \Theta (D)^u for any form u with values in E. This 2-form with values in Hom(E; E) is called the curvature tensor of the connection D. Now suppose that E is equipped with a Euclidean metric (resp. Hermitian) of class C 1 and that the isomorphism Ej \Omega ' \Omega \Theta C r is given by a C 1 frame (e*). j.-p. demailly, part i: L 2 hodge theory 8 We then have a canonical bilinear pairing, (resp. sesquilinear). C 1 (M; \Lambda p T \Lambda M \Omega E) \Theta C 1 (M; \Lambda q T \Lambda M \Omega E) ! C 1 (M; \Lambda p+q T \Lambda M \Omega C ) (1.6) (u; v) 7! fu; vg given by fu; vg = X *;_ u* ^ v_he*; e_i; u = X u* \Omega e*; v = X v_ \Omega e_: The connection D is called Hermitian if it satisfies the additional property dfu; vg = fDu; vg + (\Gamma 1) deg u fu; Dvg: By assuming that (e*) is orthonormal, one easily verifies that D is Hermitian if and only if \Gamma \Lambda = \Gamma \Gamma . In this case \Theta (D) \Lambda = \Gamma \Theta (D), therefore i\Theta (D) 2 C 1 (M; \Lambda 2 T \Lambda M \Omega Herm(E; E)): 1.7. A particular case. For a complex line bundle L (a complex vector bundle of rank 1), the connection form \Gamma of a Hermitian connection D can be taken to be a 1-form with purely imaginary coefficients \Gamma = iA (A real). We then have \Theta (D) = d\Gamma = idA. In particular i\Theta (L) is a closed 2-form. The first Chern class of L is defined to be the cohomology class c1(L)R = ae i 2ss \Theta (D) oe 2 H 2 DR (M; R): This cohomology class is independent of the choice of connection, since any other connection D1 differs by a global 1-form, D1u = Du + B ^ u, so that \Theta (D1) = \Theta (D) + dB. It is well-known that c1(L)R is the image in H 2 (M; R) of an integral class c1(L) 2 H 2 (M; Z). Indeed if A = C 1 is the sheaf of C 1 functions on M , then via the exponential exact sequence 0 ! Z ! A e 2ssiffl \Gamma \Gamma \Gamma ! A \Lambda ! 0; c1(L) can be defined in ^ Cech cohomology as the image of the cocycle fgjkg 2 H 1 (M; A \Lambda ) defining L by the coedge map H 1 (M; A \Lambda ) ! H 2 (M; Z). See for example [GH78] for more details. 1.C. Connections on complex manifolds. We now study those properties of connections governed by the existence of a complex structure on the base manifold. If M = X is a complex manifold, any connection D on a complex C 1 vector bundle E can be split in a unique manner as a sum of a (1; 0)-connection and a (0; 1)-connection, D = D 0 + D 00 . In a local trivialization o/ given by a C 1 frame, one can write D 0 u 'o/ d 0 u + \Gamma 0 ^ u;(1.8 0 ) D 00 u 'o/ d 00 u + \Gamma 00 ^ u;(1.8 00 ) with \Gamma = \Gamma 0 + \Gamma 00 . The connection is Hermitian if and only if \Gamma 0 = \Gamma (\Gamma 00 ) \Lambda relative to any orthonormal frame. As a consequence, there exists a unique Hermitian connection D associated to a (0; 1)-connection prescribed by D 00 . 9 2. differential operators on vector bundles Now suppose that the bundle E is endowed with a holomorphic structure. The unique Hermitian connection whose component D 00 is the operator d 00 defined in x1.A is called the Chern connection of E. With respect to a local holomorphic frame (e*) of Ej \Omega , the metric is given by the Hermitian matrix H = (h*_) where h*_ = he*; e_i. We have fu; vg = X *;_ h*_u* ^ v_ = u y ^ Hv; where u y is the transpose matrix of u, and an easy calculation gives dfu; vg = (du) y ^ Hv + (\Gamma 1) deg u u y ^ (dH ^ v + Hdv) = (du + H \Gamma 1 d 0 H ^ u) y ^ Hv + (\Gamma 1) deg u u y ^ (dv + H \Gamma 1 d 0 H ^ v); by using the fact that dH = d 0 H + d 0 H and H y = H. Consequently the Chern connection D coincides with the Hermitian connection defined by ( Du 'o/ du + H \Gamma 1 d 0 H ^ u; D 0 'o/ d 0 + H \Gamma 1 d 0 H ^ ffl = H \Gamma 1 d 0 (Hffl); D 00 = d 00 : (1.9) These relations show that D 02 = D 002 = 0. Consequently D 2 = D 0 D 00 + D 00 D 0 , and the curvature tensor \Theta (D) is of type (1; 1). Since d 0 d 00 + d 00 d 0 = 0, we obtain (D 0 D 00 + D 00 D 0 )u 'o/ H \Gamma 1 d 0 H ^ d 00 u + d 00 (H \Gamma 1 d 0 H ^ u) = d 00 (H \Gamma 1 d 0 H) ^ u: 1.10. Proposition. The Chern curvature tensor \Theta (E) := \Theta (D) satisfies i\Theta (E) 2 C 1 (X; \Lambda 1;1 T \Lambda X \Omega Herm(E; E)): If o/ : E_\Omega ! \Omega \Theta C r is a holomorphic trivialization and if H is the Hermitian matrix representative of the metric along the fibers of E_\Omega , then i\Theta (E) 'o/ id 00 (H \Gamma 1 d 0 H) on \Omega : \Lambda If (z1; : : : ; zn) are holomorphic coordinates on X and if (e*)1^*^r is an orthogonal frame of E, one can write (1.11) i\Theta (E) = X 1^j;k^n;1^*;_^r cjk*_dzj ^ dzk \Omega e \Lambda * \Omega e_; where (cjk*_(x)) are the coefficients of the curvature tensor of E at any point x 2 X. 2. Differential operators on vector bundles We first describe some basic concepts concerning differential operators (symbol, composition, ellipticity, adjoint), in the general context of vector bundles. Assume given M a manifold of differentiable class C 1 ; dimR M = m, and E; F given K vector bundles on M , over the field K = R or K = C such that rank E = r, rank F = r 0 . j.-p. demailly, part i: L 2 hodge theory 10 2.1. Definition. A (linear) differential operator of degree ffi from E to F is a K -linear operator P : C 1 (M; E) ! C 1 (M; F ); u 7! P u of the form P u(x) = X jffj^ffi aff(x)D ff u(x); where E_\Omega ' \Omega \Theta K r ; F_\Omega ' \Omega \Theta K r 0 are local trivializations on an open chart \Omega ae M with local coordinates (x1; : : : ; xm), and the coefficients aff(x) are r 0 \Theta r matrices (aff*_(x))1^*^r0;1^_^r with C 1 coefficients on \Omega . One writes here D ff = (@=@x1) ff1 \Delta \Delta \Delta (@=@xm) ffm as usual, and the matrices u = (u_)1^_^r, D ff u = (D ff u_)1^_^r are viewed as column vectors. If t 2 K is a parameter and f 2 C 1 (M; K ); u 2 C 1 (M; E), an easy calculation shows that e \Gamma tf(x) P (e tf(x) u(x)) is a polynomial of degree ffi in t, of the form e \Gamma tf(x) P (e tf(x) u(x)) = t ffi oeP (x; df (x)) \Delta u(x) + terms cj(x)t j of degree j ! ffi; where oeP is a homogeneous polynomial map T \Lambda M ! Hom(E; F ) defined by (2.2) T \Lambda M;x 3 , 7! oeP (x; ,) 2 Hom(Ex; Fx); oeP (x; ,) = X jffj=ffi aff(x), ff : Then oeP (x; ,) is a C 1 function of the variables (x; ,) 2 T \Lambda M , and this function is independent of the choice of coordinates or trivialization used for E; F . oeP is called the principal symbol of P . The principal symbol of a composition Q ffi P of differential operators is simply the product. (2.3) oeQffiP (x; ,) = oeQ(x; ,)oeP (x; ,); calculated as a product of matrices. The differential operators for which the symbols are injective play a very important role: 2.4. Definition. A differential operator P is said to be elliptic if oeP (x; ,) 2 Hom(Ex; Fx) is injective for all x 2 M and , 2 T \Lambda M;x nf0g. Let us now assume that M is oriented and assume given a C 1 volume form dV (x) = fl(x)dx1 ^\Delta \Delta \Delta ^dxm, where fl(x) ? 0 is a C 1 density. If E is a Euclidean or Hermitian vector bundle, we can define a Hilbert space L 2 (M; E) of global sections with values in E, being the space of forms u with measurable coefficients which are square summable sections with respect to the scalar product jjujj 2 = Z M ju(x)j 2 dV (x);(2.5) hhu; vii = Z M hu(x); v(x)idV (x); u; v 2 L 2 (M; E):(2.5 0 ) 2.6. Definition. If P : C 1 (M; E) ! C 1 (M; F ) is a differential operator and if the bundles E; F are Euclidean or Hermitian, there exists a unique differential operator P \Lambda : C 1 (M; F ) ! C 1 (M; E); called the formal adjoint of P , such that for all sections u 2 C 1 (M; E) and v 2 C 1 (M; F ) one has an identity hhP u; vii = hhu; P \Lambda vii; whenever Supp u " Supp v aeae M: 11 3. fundamental results on elliptic operators Proof. The uniqueness is easy to verify, being a consequence of the density of C 1 forms with compact support in L 2 (M; E). By a partition of unity argument, we reduce the verification of the existence of P \Lambda to the proof of its local existence. Now let P u(x) = \Sigma jffj^ffiaff(x)D ff u(x) be the description of P relative to the trivializations of E; F associated to an orthonormal frame and to the system of local coordinates on an open set \Omega ae M . By assuming Supp u " Supp v aeae \Omega , integration by parts gives hhP u; vii = Z \Omega X jffj^ffi;*;_ aff*_D ff u_(x)v*(x)fl(x)dx1; : : : ; dxm = Z \Omega X jffj^ffi;*;_ (\Gamma 1) jffj u_(x)D ff (fl(x)aff*_v*(x)dx1; : : : ; dxm = Z \Omega hu; X jffj^ffi (\Gamma 1) jffj fl(x) \Gamma 1 D ff \Gamma fl(x)a y ff v(x) \Delta idV (x): We thus see that P \Lambda exists, and is defined in a unique way by (2.7) P \Lambda v(x) = X jffj^ffi (\Gamma 1) jffj fl(x) \Gamma 1 D ff \Gamma fl(x)a y ff v(x) \Delta : \Lambda Formula (2.7) shows immediately that the principal symbol of P \Lambda is given by (2.8) oeP \Lambda (x; ,) = (\Gamma 1) ffi X jffj=ffi a y ff , ff = (\Gamma 1) ffi oeP (x; ,) \Lambda : If rank E = rank F , the operator P is elliptic if and only if oeP (x; ,) is invertible for , 6= 0, therefore the ellipticity of P is equivalent to that of P \Lambda . 3. Fundamental results on elliptic operators We assume throughout this section that M is a compact oriented C 1 manifold of dimension m, with volume form dV . Let E ! M be a C 1 Hermitian vector bundle of rank r on M . 3.A. Sobolev spaces. For any real number s, we define the Sobolev space W s (R m ) to be the Hilbert space of tempered distributions u 2 S 0 (R m ) such that the Fourier transform ^u is a L 2 loc function satisfying the estimate (3.1) jjujj 2 s = Z Rm (1 + j,j 2 ) s j^u(,)j 2 d*(,) ! +1: If s 2 N, we have jjuj 2 s , Z Rm X jffj^s jD ff u(x)j 2 d*(x); therefore W s (R m ) is the Hilbert space of functions u such that all the derivatives D ff u of order jffj ^ s are in L 2 (R m ). More generally, we denote by W s (M; E) the Sobolev space of sections u : M ! E whose components are locally in W s (R m ) on all open charts. More precisely, choose a finite subcovering (\Omega j ) of M by open coordinate charts \Omega j ' R m on which E is trivial. j.-p. demailly, part i: L 2 hodge theory 12 Consider an orthonormal frame (ej;*)1^*^r of E_\Omega j and write u in terms of its components, i.e. u = P uj;*ej;*. We then set jjujj 2 s = X j;* jjjuj;*jj 2 s where (j) is a "partition of unity" subordinate to (\Omega j), such that P 2 j = 1. The equivalence of norms jj jjs is independent of choices made. We will need the following fundamental facts, that the reader will be able to find in many of the specialized works devoted to the theory of partial differential equations. 3.2. Sobolev lemma. For an integer k 2 N and any real numbers s * k + m 2 , we have W s (M; E) ae C k (M; E) and the inclusion is continuous. \Lambda It follows immediately from the Sololev lemma that " s*0 W s (M; E) = C 1 (M; E); [ s^0 W s (M; E) = D 0 (M; E): 3.3. Rellich lemma. For all t ? s, the inclusion W t (M; E) ,! W s (M; E) is a compact linear operator. \Lambda 3.B. Pseudodifferential operators. If P = P jffj^ffi aff(x)D ff is a differential operator on R m , the Fourier inversion formula gives P u(x) = Z Rm X jffj^ffi aff(x)(2ssi,) ff ^u(,)e 2ssix\Delta , d*(,); 8u 2 D(R m ); where ^u(,) = R Rm u(x)e \Gamma 2ssix\Delta , d*(x) is the Fourier transform of u. We call oe(x; ,) = X jffj^ffi aff(x)(2ssi,) ff ; the symbol (or total symbol) of P . A pseudodifferential operator is an operator Opoe defined by a formula of the type (3.4) Opoe(u)(x) = Z Rm oe(x; ,)^u(,)e 2ssix\Delta , d*(,); u 2 D(R m ); where oe belongs to a suitable class of functions on T \Lambda Rm . The standard class of symbols S ffi (R m ) is defined as follows: Assume given ffi 2 R; S ffi (R m ) is the class of C 1 functions oe(x; ,) on T \Lambda Rm such that for any ff; fi 2 N m and any compact subset K ae R m one has an estimate (3.5) jD ff x D fi , oe(x; ,)j ^ Cff;fi(1 + j,j) ffi\Gamma jfij ; 8(x; ,) 2 K \Theta R m ; where ffi 2 R is regarded as the "degree" of oe. Then Opoe(u) is a well defined C 1 function on R m , since ^u belongs to the class S(R m ) of functions having rapid decay. In the more general situation of operators acting on a bundle E and having values 13 3. fundamental results on elliptic operators in a bundle F over a compact manifold M , we introduce the analogous space of symbols S ffi (M ; E; F ). The elements of S ffi (M ; E; F ) are the functions T \Lambda M 3 (x; ,) 7! oe(x; ,) 2 Hom(Ex; Fx) satisfying condition (3.5) in all coordinate systems. Finally, we take a finite trivializing cover (\Omega j) of M and a "partition of unity" (j) subordinate to \Omega j such that P 2 j = 1, and we define Opoe(u) = X jOpoe(ju); u 2 C 1 (M; E); in a way which reduces the calculations to the situation of R m . The basic results pertaining to the theory of pseudodifferential operators are summarized below. 3.6. Existence of extensions to the spaces W s . If oe 2 S ffi (M ; E; F ), then Opoe extends uniquely to a continuous linear operator Opoe : W s (M; E) ! W s\Gamma ffi (M; F ): \Lambda In particular if oe 2 S \Gamma 1 (M ; E; F ) := T S ffi (M ; E; F ), then Opoe is a continuous operator sending an arbitrary distributional section of D 0 (M; E) into C 1 (M; F ). Such an operator is called a regular operator. It is a standard result in the theory of distributions that the class R of regular operators coincides with the class of operators defined by means of a C 1 kernel K(x; y) 2 Hom(Ey; Fx). That is, the operators of the form R : D 0 (M; E) ! C 1 (M; F ); u 7! Ru; Ru(x) = Z M K(x; y) \Delta u(y)dV (y): Conversely, if dV (y) = fl(y)dy1 \Delta \Delta \Delta dym on \Omega j and if we write Ru = P R(`ju), where (`j) is a partition of unity, the operator R(`jffl) is the pseudodifferential operator associated to the symbol oe defined by the partial Fourier transform oe(x; ,) = \Gamma fl(y)`j(y)K(x; y) \Delta ^ y (x; ,); oe 2 S \Gamma 1 (M ; E; F ): When one works with pseudodifferential operators, it is customary to work modulo the regular operators and to allow operators more generally of the form Opoe + R where R 2 R is an arbitrary regular operator. 3.7. Composition. If oe 2 S ffi (M ; E; F ) and oe 0 2 S ffi 0 (M ; F; G); ffi; ffi 0 2 R, there exists a symbol oe 0 ""oe 2 S ffi+ffi 0 (M ; E; G) such that Opoe0 ffi Opoe = Opoe0""oe mod R. Moreover oe 0 ""oe \Gamma oe 0 \Delta oe 2 S ffi+ffi 0 \Gamma 1 (M ; E; G): 3.8. Definition. A pseudodifferential operator Opoe of degree ffi is called elliptic if it can be defined by a symbol oe 2 S ffi (M; E; F ) such that joe(x; ,) \Delta uj * cj,j ffi juj; 8(x; ,) 2 T \Lambda M ; 8u 2 Ex for j,j large enough, the estimate being uniform for x 2 M . If E and F have the same rank, the ellipticity condition implies that oe(x; ,) is invertible for large ,. By taking a suitable truncating function `(,) equal to 1 for large ,, one sees that the function oe 0 (x; ,) = `(,)oe(x; ,) \Gamma 1 defines a symbol in the space S \Gamma ffi (M ; F; E), and according to (3.8) we have Opoe0 ffi Opoe = Id + Opae; ae 2 S \Gamma 1 (M ; E; E). Choose a symbol o/ asymptomatically equivalent (at infinity) to the j.-p. demailly, part i: L 2 hodge theory 14 expansion Id \Gamma ae + ae ""2 + \Delta \Delta \Delta + (\Gamma 1) j ae ""j + \Delta \Delta \Delta . It is clear then that one obtains an inverse Opo/""oe0 of Opoe modulo R. An easy consequence of this observation is the following: 3.9. G*arding inequality. Assume given P : C 1 (M; E) ! C 1 (M; F ) an elliptic differential operator of degree ffi, where rank E = rank F = r, and let ~ P be an extension of P with distributional coefficient sections. For all u 2 W 0 (M; E) such that ~ P u 2 W s (M; F ), one then has u 2 W s+ffi (M; E) and jjujjs+ffi ^ Cs(jj ~ P ujjs + jjujj0); where Cs is a positive constant depending only on s. Proof. Since P is elliptic, there exists a symbol oe 2 S \Gamma ffi (M ; F; E) such that Opoe ffi ~ P = Id + R; R 2 R. Then jjOpoe(v)jjs+ffi ^ Cjjvjjs by applying (3.6). Consequently, in setting v = ~ P u, we see that u = Opoe( ~ P u) \Gamma Ru satisfies the desired estimate. \Lambda 3.C. Finiteness theorem. We conclude this section with the proof of the following fundamental finiteness theorem, which is the starting point of L 2 Hodge theory. 3.10. Finiteness Theorem. Assume given E; F Hermitian vector bundles on a compact manifold M , such that rank E = rank F = r; and given P : C 1 (M; E) ! C 1 (M; F ) an elliptic differential operator of degree ffi. Then: i) Ker P is finite dimensional. ii) P (C 1 (M; E)) is closed and of finite codimension in C 1 (M; F ); moreover, if P \Lambda is the formal adjoint of P , there exists a decomposition. C 1 (M; F ) = P (C 1 (M; E)) \Phi Ker P \Lambda as an orthogonal direct sum in W 0 (M; F ) = L 2 (M; F ). Proof. (i) The G*arding inequality shows that jjujjs+ffi ^ Csjjujj0 for all u 2 Ker P . By the Sobolev Lemma, this implies that Ker P is closed in W 0 (M; E). Moreover, the jj jj0-closed unit ball of Ker P is contained in the jj jjffi-ball of radius C0, therefore it is compact according to the Rellich Lemma. Riesz Theorem implies that dim Ker P ! +1. (ii) We first show that the extension ~ P : W s+ffi (M; E) ! W s (M; F ) has closed image for all s. For any ffl ? 0, there exists a finite number of elements v1; : : : ; vN 2 W s+ffi (M; F ); N = N (ffl), such that (3.11) jjujj0 ^ ffljjujjs+ffi + N X j=1 jhhu; vj ii0j: Indeed the set: K(v j) = ae u 2 W s+ffi (M; F ) ; ffljjujjs+ffi + N X j=1 jhhu; vjii0j ^ 1 oe ; 15 3. fundamental results on elliptic operators is relatively compact in W 0 (M; F ) and T (vj) K(v j) = f0g. It follows that there are elements (vj ) such that K(v j) are contained in the unit ball of W 0 (M; E), as required. Substituting the main term jjujj0 given by (3.11) in the G*arding inequality; we obtain (1 \Gamma Csffl)jjujjs+ffi ^ Cs ` jj ~ P ujjs + N X j=1 jhhu; vj ii0j ' : Define T = fu 2 W s+ffi (M; E) ; u ? vj; 1 ^ j ^ ng and put ffl = 1=2Cs. It follows that jjujjs+ffi ^ 2Csjj ~ P ujjs; 8u 2 T: This implies that ~ P (T ) is closed. As a consequence ~ P (W s+ffi (M; E)) = ~ P (T ) + Vect \Gamma ~ P (v1); : : : ; ~ P (vN ) \Delta is closed in W s (M; E). Consider now the case s = 0. Since C 1 (M; E) is dense in W ffi (M; E), we see that in W 0 (M; E) = L 2 (M; E), one has ` ~ P \Gamma W ffi (M; E) \Delta '? = ` P \Gamma C 1 (M; E) \Delta '? = Ker ~ P \Lambda : We have thus proven that (3.12) W 0 (M; E) = ~ P \Gamma W ffi (M; E) \Delta \Phi Ker ~ P \Lambda : Since P \Lambda is also elliptic, it follows that Ker ~ P \Lambda is finite dimensional and that Ker ~ P \Lambda = Ker P \Lambda is contained in C 1 (M; F ). By applying the G*arding inequality, the decomposition formula (3.12) gives W s (M; E) = ~ P \Gamma W s+ffi (M; E) \Delta \Phi Ker P \Lambda ;(3.13) C 1 (M; E) = P \Gamma C 1 (M; E) \Delta \Phi Ker P \Lambda :(3.14) We finish this section by the construction of the Green's operator associated to a self-adjoint elliptic operator. 3.15. Theorem. Assume given E a Hermitian vector bundle of rank r on a compact manifold M , and P : C 1 (M; E) ! C 1 (M; E) a self-adjoint elliptic differential operator of degree ffi. Then if H denotes the orthogonal projection operator H : C 1 (M; E) ! Ker P , there exists a unique operator G on C 1 (M; E) such that P G + H = GP + H = Id; moreover G is a pseudo-differential operator of degree \Gamma ffi, called the Green's operator associated to P . Proof. According to Theorem 3.10, Ker P = Ker P \Lambda is finite dimensional and Im P = (Ker P ) ? . It then follows that the restriction of P to (Ker P ) ? is a bijective operator. One defines G to be 0 \Phi P \Gamma 1 relative to the orthogonal decomposition C 1 (M; E) = Ker P \Phi (Ker P ) ? . The relations P G + H = GP + H = Id are then obvious, as well as the uniqueness of G. Moreover, G is continuous in the Fr'echet space topology of C 1 (M; E), according to the Banach theorem. One also uses j.-p. demailly, part i: L 2 hodge theory 16 the fact that there exists a pseudo-differential operator Q of order \Gamma ffi which is an inverse of P modulo R, i.e. P Q = Id + R; R 2 R. It then follows that Q = (GP + H)Q = G(Id + R) + HQ = G + GR + HQ; where GR and HG are regular. (H is a regular operator of finite rank defined by the kernel P 's(x) \Omega ' \Lambda s (y), if ('s) is a basis of eigenfunctions of Ker P ae C 1 (M; E).) Consequently G = Q mod R and G is a pseudodifferential operator of order \Gamma ffi. \Lambda 3.16. Corollary. Under the hypotheses of 3.15, the eigenvalues of P form a real sequence *k such that limk!+1 j*kj = +1, the eigenspaces V* k of P are finite dimensional, and one has a Hilbert space direct sum L 2 (M; E) = d M k V* k : For any integer m 2 N, an element u = P k uk 2 L 2 (M; E) is in W mffi (X; E) if and only if P j*kj 2m jjukjj 2 ! +1. Proof. The Green's operator extends to a self-adjoint operator ~ G : L 2 (M; E) ! L 2 (M; E) which factors through W ffi (M; E), and is therefore compact. This operator defines an inverse to ~ P : W ffi (M; E) ! L 2 (M; E) on (Ker P ) ? . The spectral theory of compact self-adjoint operators shows that the eigenvalues _k of ~ G form a real sequence tending to 0 and that L 2 (M; E) is a direct sum of Hilbert eigenspaces. The corresponding eigenvalues of ~ P are *k = _ \Gamma 1 k if _k 6= 0 and according to the ellipticity of P \Gamma *kId, the eigenspaces V* k = Ker(P \Gamma *kId) are finite dimensional and contained in C 1 (M; E). Finally, if u = P k uk 2 L 2 (M; E), the G*arding inequality shows that u 2 W mffi (M; E) if and only if ~ P m u 2 L 2 (M; E) = W 0 (M; E), which easily gives the condition P j*kj 2m jjukjj 2 ! +1. 4. Hodge theory of compact Riemannian manifolds The establishment of Hodge theory as a well developed subject, was carried out by W.V.D Hodge during the decade 1930-1940 (see [Hod41], [DR55]). The principal goal of the theory is to describe the de Rham cohomology algebra of a Riemannian manifold in terms of its harmonic forms. The principal result is that any cohomology class has a unique harmonic representative. 4.A. Euclidean structure of the exterior algebra. Let (M; g) be an oriented Riemannian C 1 manifold of dimension m, and let E ! M be a Hermitian vector bundle of rank r on M . We denote respectively by (,1; : : : ; ,m) and (e1; : : : ; er) orthonormal frames of TM and of E on a coordinate chart \Omega ae M , and let (, \Lambda 1 ; : : : ; , \Lambda m ), (e \Lambda 1 ; : : : ; e \Lambda r ) be the corresponding dual coframes of T \Lambda M ; E \Lambda respectively. Further, let dV be the Riemannian volume element on M . The exterior algebra \Lambda ffl T \Lambda M is endowed with a natural inner product hffl; ffli, given by (4.1) hu1 ^ \Delta \Delta \Delta ^ up; v1 ^ \Delta \Delta \Delta ^ vpi = det(huj; vki)1^j;k^p; uj; vk 2 T \Lambda M for all p, with \Lambda ffl T \Lambda M = L \Lambda p T \Lambda M an orthogonal direct sum. Thus the family of covectors , \Lambda I = , \Lambda i1 ^ \Delta \Delta \Delta ^ , \Lambda ip ; i1 ! i2 ! \Delta \Delta \Delta ! ip, defines an orthonormal basis of \Lambda ffl T \Lambda M . One denotes by hffl; ffli the corresponding inner product on \Lambda ffl T \Lambda M \Omega E. 17 4. hodge theory of compact Riemannian manifolds 4.2. Hodge star operator. The Hodge-Poincar'e-de Rham ? operator is the endomorphism of \Lambda ffl T \Lambda M defined by a collection of linear maps such that ? : \Lambda p T \Lambda M ! \Lambda m\Gamma p T \Lambda M ; u ^ ?v = hu; vidV; 8u; v 2 \Lambda p T \Lambda M : The existence and uniqueness of this operator follows easily from the duality pairing \Lambda p T \Lambda M \Theta \Lambda m\Gamma p T \Lambda M ! R (4.3) (u; v) 7! u ^ v=dV = X ffl(I; -I)uIv-I ; where u = P jIj=p uI , \Lambda I ; v = P jJj=m\Gamma p vJ , \Lambda J , and where ffl(I; -I) is the sign of the permutation (1; 2; : : : ; m) 7! (I; -I) defined by I followed by the complementary (ordered) multi-indices -I. From this, we deduce (4.4) ?v = X jIj=p ffl(I; -I)vI , \Lambda -I : More generally, the sesquilinear pairing fffl; fflg defined by (1.6) induces an operator ? on the vector-valued forms, such that (4.5) ? : \Lambda p T \Lambda M \Omega E ! \Lambda m\Gamma p T \Lambda M \Omega E; fs; ?tg = hs; tidV; (4.6) ?t = X jIj=p;* ffl(I; -I)tI;*, \Lambda -I \Omega e*; 8s; t 2 \Lambda p T \Lambda M \Omega E; for t = P tI;*, \Lambda I \Omega e*. Since ffl(I; -I)ffl(-I; I) = (\Gamma 1) p(m\Gamma p) = (\Gamma 1) p(m\Gamma 1) , we immediately obtain (4.7) ? ? t = (\Gamma 1) p(m\Gamma 1) t on \Lambda p T \Lambda M \Omega E: It is clear that ? is an isometry of \Lambda ffl T \Lambda M \Omega E. We will also need a variant of the ? operator, namely the antilinear operator # : \Lambda p T \Lambda M \Omega E ! \Lambda m\Gamma p T \Lambda M \Omega E \Lambda defined by s ^ #t = hs; tidV , where the exterior product ^ is combined with the canonical pairing E \Theta E \Lambda ! C . We have (4.8) #t = X jIj=p;* ffl(I; -I)tI;*, \Lambda -I \Omega e \Lambda * : 4.9. Contraction by a vector field. Assume given a tangent vector ` 2 TM and a form u 2 \Lambda p T \Lambda M . The contraction `yu 2 \Lambda p\Gamma 1 T \Lambda M is defined by `yu(j1; : : : ; jp\Gamma 1) = u(`; j1; : : : ; jp\Gamma 1); jj 2 TM : In terms of the basis (,j ); fflyffl is the bilinear operator characterized by ,ly(, \Lambda i1 ^ \Delta \Delta \Delta ^ , \Lambda ip ) = ( 0 if l 62 fi1; : : : ; ipg, (\Gamma 1) k\Gamma 1 , \Lambda i1 ^ \Delta \Delta \Delta ^ , \Lambda ik \Delta \Delta \Delta , \Lambda ip if l = ik: This same formula is also valid when (,j ) is not orthonormal. An easy calculation shows that `yffl is a derivation of the exterior algebra, i.e. that `y(u ^ v) = (`yu) ^ v + (\Gamma 1) deg u u ^ (`yv): j.-p. demailly, part i: L 2 hodge theory 18 Moreover, if ~ ` = hffl; `i 2 T \Lambda M , the operator `yffl is the adjoint of ~ ` ^ ffl, i.e., (4.10) h`yu; vi = hu; ~ ` ^ vi; 8u; v 2 \Lambda ffl T \Lambda M : Indeed, this property is immediate when ` = ,l; u = , \Lambda I ; v = , \Lambda J . 4.B. Laplace-Beltrami operator. Let E be a Hermitian vector bundle on M , and let DE be a Hermitian connection on E. We consider the Hilbert space L 2 (M; \Lambda p T \Lambda M \Omega E) of p-forms on M with values in E, with the given L 2 scalar product hhs; tii = Z M hs; tidV already introduced in (2.5). Here hs; ti is the specific scalar product on \Lambda p T \Lambda M \Omega E associated to the Riemannian scalar product on \Lambda p T \Lambda M and the Hermitian pairing on E. 4.11. Theorem. The formal adjoint of DE acting on C 1 (M; \Lambda p T \Lambda M \Omega E) is given by D \Lambda E = (\Gamma 1) mp+1 ? DE ? : Proof. If s 2 C 1 (M; \Lambda p T \Lambda M \Omega E) and t 2 C 1 (M; \Lambda p+1 T \Lambda M \Omega E) have compact support, we have hhDEs; tii = Z M hDEs; tidV = Z M fDEs; ?tg = Z M dfs; ?tg \Gamma (\Gamma 1) p fs; DE ? tg = (\Gamma 1) p+1 Z M fs; DE ? tg by an application of Stokes theorem. As a consequence, (4.5) and (4.7) imply hhDEs; tii = (\Gamma 1) p+1 (\Gamma 1) p(m\Gamma 1) Z M fs; ? ? DE ? tg = (\Gamma 1) mp+1 hhs; ?DE ? tii: The desired formula follows. \Lambda 4.12. Remark. In the case of the trivial connection d on E = M \Theta C , the formula becomes d \Lambda = (\Gamma 1) m+1 ? d?. If m is even, these formulas reduce to d \Lambda = \Gamma ? d?; D \Lambda E = \Gamma ? DE ? : 4.13. Definition. The Laplace-Beltrami operator is the second order differential operator acting on the bundle, \Lambda p T \Lambda M \Omega E, such that \Delta E = DED \Lambda E + D \Lambda E DE: In particular, the Laplace-Beltrami operator acting on \Lambda p T \Lambda M is \Delta = dd \Lambda + d \Lambda d. This latter operator does not depend on the Riemannian structure (M; g). It is clear that the Laplacian \Delta is formally self-adjoint i.e. hh\Delta Es; tii = hhs; \Delta Etii whenever the forms s; t are C 1 and that one of them has compact support. 19 4. hodge theory of compact Riemannian manifolds 4.14. Calculation of the symbol. For every C 1 function f , Leibnitz rule gives e \Gamma tf DE(e tf s) = tdf ^ s + DEs. By definition of the symbol, we therefore find oeD E (x; ,) \Delta s = , ^ s; 8, 2 T \Lambda M;x ; 8s 2 \Lambda p T \Lambda M \Omega E: From formula (2.8), we obtain oeD\Lambda E = \Gamma (oeD E ) \Lambda , therefore oeD\Lambda E (x; ,) \Delta s = \Gamma ~ ,ys where ~ , 2 TM is the adjoint tangent vector of ,. The equality oe\Delta E = oeD E oeD\Lambda E + oeD\Lambda E oeD E implies that oe\Delta E (x; ,) \Delta s = \Gamma , ^ ( ~ ,ys) \Gamma ~ ,y(, ^ s) = \Gamma ( ~ ,y,)s; oe\Delta E (x; ,) \Delta s = \Gamma j,j 2 s: In particular, \Delta E is always an elliptic operator. In the special case where M is an open subset of R m with the constant metric g = P m i=1 dx 2 i , all these operators d; d \Lambda ; \Delta have constant coefficients. They are completely determined by their principal symbol (no term of lower order can appear). One easily computes: s = X jIj=p sIdxI ; ds = X jIj=p;j @sI @xj dxj ^ dxI ; d \Lambda s = \Gamma X I;j @sI @xj @ @xj ydxI; \Delta s = \Gamma X I ` X j @ 2 sI @x 2 j ' dxI : Consequently \Delta has the same expression as the elementary Laplacian operator, up to a minus sign. 4.C. Harmonic forms and the Hodge isomorphism. Let E be a Hermitian vector bundle on a compact Riemannian manifold (M; g). We assume that E is given a Hermitian connection DE such that \Theta (DE) = D 2 E = 0. Such a connection is said to be integrable or flat. It is known that this is equivalent to such an E given by a representation ss1(M ) ! U (r). Such a bundle is called a flat bundle or a local system of coefficients. A standard example is the trivial bundle E = M \Theta C with its obvious connection DE = d. Our assumption implies that DE defines a generalized de Rham complex C 1 (M; E) DE \Gamma \Gamma ! C 1 (M; \Lambda 1 T \Lambda M \Omega E) ! \Delta \Delta \Delta ! C 1 (M; \Lambda p T \Lambda M \Omega E) DE \Gamma \Gamma ! \Delta \Delta \Delta : The cohomology groups of this complex are denoted by H p DR (M; E). The space of harmonic forms of degree p relative to the Laplace-Beltrami operator \Delta E = DED \Lambda E + D \Lambda E DE is defined by H p (M; E) = \Phi s 2 C 1 (M; \Lambda p T \Lambda M \Omega E) ; \Delta Es = 0 \Psi : Since hh\Delta Es; sii = jjDesjj 2 + jjD \Lambda E sjj 2 , we see that s 2 H p (M; E) if and only if DEs = D \Lambda E s = 0. j.-p. demailly, part i: L 2 hodge theory 20 4.16. Theorem. For all p, there exists an orthogonal decomposition C 1 (M; \Lambda p T \Lambda M \Omega E) = H p (M; E) \Phi ImDE \Phi ImD \Lambda E ; where ImDE = DE \Gamma C 1 (M; \Lambda p\Gamma 1 T \Lambda M \Omega E) \Delta ; ImD \Lambda E = D \Lambda E \Gamma C 1 (M; \Lambda p+1 T \Lambda M \Omega E) \Delta : Proof. It is immediate that H p (M; E) is orthogonal to the two subspaces ImDE and ImD \Lambda E . The orthogonality of these two subspaces is also obvious, as a result of the hypothesis D 2 E = 0, namely: hhDEs; D \Lambda E tii = hhD 2 E s; tii = 0: We now apply th. 3.10 to the elliptic operator \Delta E = \Delta \Lambda E acting on the p-forms, i.e. the operator \Delta E : C 1 (M; F ) ! C 1 (M; F ) acting on the bundle F = \Lambda p T \Lambda M \Omega E. We obtain C 1 (M; \Lambda p T \Lambda M \Omega E) = H p (M; E) \Phi \Delta E(C 1 (M; \Lambda p T \Lambda M \Omega E)); Im\Delta E = Im(DED \Lambda E + D \Lambda E DE) ae ImDE + ImD \Lambda E : Further, since ImDE and ImD \Lambda E are orthogonal to H p (M; E), these spaces are contained in Im\Delta E. \Lambda 4.17. Hodge Isomorphism Theorem. The de Rham cohomology groups H p DR (M; E) are finite dimensional; moreover H p DR (M; E) ' H p (M; E). Proof. From the decomposition in (4.16), we obtain B p DR (M; E) = DE(C 1 (M; \Lambda p\Gamma 1 T \Lambda M \Omega E)); Z p DR (M; E) = Ker DE = (ImD \Lambda E ) ? = H p (M; E) \Phi ImDE: This shows that any de Rham cohomology class contains a unique harmonic representative. \Lambda 4.18. Poincar'e duality. The pairing H p DR (M; E) \Theta H m\Gamma p DR (M; E \Lambda ) ! C ; (s; t) 7! Z M s ^ t is a non-degenerate bilinear form, and thus defines a duality between H p DR (M; E) and H m\Gamma p DR (M; E \Lambda ): Proof. First observe that there is a naturally defined flat connection DE\Lambda such that for all s 2 C 1 (M; \Lambda ffl T \Lambda M \Omega E); t 2 C 1 (M; \Lambda ffl T \Lambda M \Omega E \Lambda ), one has (4.19) d(s ^ t) = (DEs) ^ t + (\Gamma 1) deg s s ^ DE\Lambda t: It then follows from Stokes theorem that the bilinear map (s; t) 7! R M s ^ t factors through the cohomology groups. For s 2 C 1 (M; \Lambda p T \Lambda M \Omega E), the reader can easily verify the following formulas (use (4.19) in a similar way to that which was done for the proof of th. 4.11): (4.20) DE\Lambda (#s) = (\Gamma 1) p #D \Lambda E s; (DE\Lambda ) \Lambda (#s) = (\Gamma 1) p+1 #DEs; \Delta E\Lambda (#s) = #\Delta s E : 21 5. Hermitian and k"ahler manifolds Consequently #s 2 H m\Gamma p (M; E \Lambda ) if and only if s 2 H p (M; E). Since Z M s ^ #s = Z M jsj 2 dV = jjsjj 2 ; it follows that the Poincar'e duality pairing has trivial kernel in the left factor H p (M; E) ' H p DR (M; E). By symmetry, it also has trivial kernel in the right. This completes the proof. \Lambda 5. Hermitian and K"ahler manifolds Let X be a complex manifold of dimension n. A Hermitian metric on X is a positive definite Hermitian C 1 form on TX . In terms of local coordinates (z1; : : : ; zn), such a form can be written h(z) = X 1^j;k^n hjk(z)dzj \Omega dzk; where (hjk) is a positive Hermitian matrix with C 1 coefficients. The fundamental (1; 1)-form associated to h is ! = \Gamma Imh = i 2 X hjkdzj ^ dzk; 1 ^ j; k ^ n: 5.1. Definition. a) A Hermitian manifold is a pair (X; !) where ! is a positive definite C 1 (1; 1)- form on X. b) The metric ! is said to be K"ahler if d! = 0. c) X is called a K"ahler manifold if X has at least one K"ahler metric. Since ! is real, the conditions d! = 0; d 0 ! = 0; d 00 ! = 0 are all equivalent. In local coordinates, we see that d 0 ! = 0 if and only if @hjk @zl = @hlk @zj ; 1 ^ j; k; l ^ n: A simple calculation gives ! n n! = det(hjk) ^ 1^j^n ` i 2 dzj ^ dzj ' = det(hjk)dx1 ^ dy1 ^ \Delta \Delta \Delta ^ dxn ^ dyn; where zn = xn + iyn. Consequently the (n; n) form (5.2) dV = 1 n! ! n is positive and coincides with the Hermitian volume element of X. If X is compact, then R X ! n = n!Vol!(X) ? 0. This simple observation already implies that a compact K"ahler manifold must satisfy certain restrictive topological conditions: 5.3. Consequence. a) If (X; !) is compact K"ahler and if f!g denotes the cohomology class of ! in H 2 (X; R), then f!g n 6= 0. b) If X is compact K"ahler, then H 2k (X; R) 6= 0 for 0 ^ k ^ n. Indeed, f!g k is a non-zero class of H 2k (X; R). j.-p. demailly, part i: L 2 hodge theory 22 5.4. Example. Complex projective space P n is endowed with a natural K"ahler metric !, called the Fubini-Study metric, defined by p \Lambda ! = i 2ss d 0 d 00 log(jioj 2 + ji1j 2 + \Delta \Delta \Delta + jinj 2 ) where i0; i1; : : : ; in are coordinates of C n+1 and where p : C n+1 nf0g ! P n is the projection. Let z = (i1=i0; : : : ; in=i0) be the non-homogeneous coordinates of the chart C n ae P n . A calculation shows that ! = i 2ss d 0 d 00 log(1 + jzj 2 ) = i 2ss \Theta (O(1)); Z Pn ! n = 1: Since the only non-zero integral cohomology groups of P n are H 2p (P n ; Z) ' Z for 0 ^ p ^ n, we see that h = f!g 2 H 2 (P n ; Z) is a generator of the cohomology ring H ffl (P n ; Z). In other words, H ffl (P n ; Z) ' Z[h]=(h n+1 ) as rings. 5.5. Example. A complex torus is a quotient X = C n =\Gamma of C n by a lattice \Gamma of rank 2n. This gives a compact complex manifold. Any positive definite Hermitian form ! = i P hjkdzj ^ dzk with constant coefficients on C n defines a K"ahler metric on X. 5.6. Example. Any complex submanifold X of a K"ahler manifold (Y; ! 0 ) is K"ahler with the induced metric ! = ! 0 _X . In particular, any projective manifold is K"ahler (by definition, a projective manifold is a closed submanifold X ae P n of projective space). In this case, if ! 0 denotes the Fubini-Study metric on P n , we have the additional property that the class f!g := f! 0 g_X 2 H 2 DR (X; R) is integral, i.e. is the image of an integral class of H 2 (X; Z). A K"ahler metric ! with integral cohomology class is called a Hodge metric. 5.7. Example. Consider the complex surface X = (C 2 nf0g)=\Gamma where \Gamma = f* n ; n 2 Zg, * 2 ]0; 1[, is viewed as a group of dilations. Since C 2 nf0g is diffeomorphic to R \Lambda + \Theta S 3 , we have X ' S 1 \Theta S 3 . As a consequence, H 2 (X; R) = 0 by an application of the K"unneth formula, and property 5.3 b) shows that X is not K"ahler. More generally, one can take for \Gamma an infinite cyclic group generated by the holomorphic contractions of C 2 , of the form ` z1 z2 ' 7! ` *1z1 *2z2 ' ; resp. ` z1 z2 ' 7! ` *z1 *z2 + z p 1 ' ; where *; *1; *2 are complex numbers such that 0 ! j*1j ^ j*2j ! 1; 0 ! j*j ! 1, and p a positive integer. These non-K"ahler surfaces are called Hopf surfaces. \Lambda The following theorem shows that a Hermitian metric ! on X is K"ahler if and only if the metric ! is tangent to order 2 to a Hermitian metric with constant coefficients at any point of X. 5.8. Theorem. Let ! be a positive definite C 1 (1; 1)-form on X. For ! to be K"ahler, it is necessary and sufficient to show that at any point x0 2 X, there exists a holomorphic coordinate system (z1; : : : ; zn) centered at x0 such that (5.9) ! = i X 1^l;m^n !lmdzl ^ dzm; !lm = ffilm + O(jzj 2 ): 23 5. Hermitian and k"ahler manifolds If ! is K"ahler, the coordinates (zj)1^j^n can be chosen so that (5.10) !lm = h @ @zl ; @ @zm i = ffilm \Gamma X 1^j;k^n cjklmzjzk + O(jzj 3 ); where (cjklm) are the coefficients of the Chern curvature tensor (5.11) \Theta (TX)x 0 = X j;k;l;m cjklmdzj ^ dzk \Omega ` @ @zl '\Lambda \Omega @ @zm associated to (TX ; !) at x0. Such a system (zj) is called a geodesic coordinate system at x0. Proof. It is clear that (5.9) implies dx 0 ! = 0, consequently the condition is sufficient. Assume now that ! is K"ahler. Then one can choose local coordinates (i1; : : : ; in) such that (di1; : : : ; din) are a !-orthonormal basis of T \Lambda x0 X. As a consequence ! = i X 1^l;m^n ~!lmdil ^ di m ; where ~!lm = ffilm + O(jij) = ffilm + X 1^j^n (ajlmij + a 0 jlm i j ) + O(jij 2 ): (5.12) Since ! is real, we have a 0 jlm = ajml. Furthermore, the K"ahler condition @!lm=@ij = @!jm=@il at x0 implies that ajlm = aljm. Now put zm = im + 1 2 X j;l ajlmij il; 1 ^ m ^ n: Then (zm) is a local coordinate system at x0, and dzm = dim + X j;l ajlmijdil; i X m dzm ^ dzm = i X m dim ^ di m + i X j;l;m ajlmijdil ^ di m + i X j;l;m ajlmi j dim ^ di l + O(jij 2 ) = i X l;m ~!lmdil ^ di m + O(jij 2 ) = ! + O(jzj 2 ): Thus we have shown condition (5.9). Now let us assume the coordinates (im) were chosen initially so that (5.9) is satisfied for (im). By continuing the Taylor expansion (5.12) to order two, we arrive at ~!lm = ffilm + O(jij 2 )) = ffilm + X jk (ajklmiji k + a 0 jklm ij ik + a 00 jklm i j i k ) + O(jij 3 ):(5.13) The new coefficients introduced satisfy the relation a 0 jklm = a 0 kjlm ; a 00 jklm = a 0 jkml ; ajklm = akjml: j.-p. demailly, part i: L 2 hodge theory 24 The K"ahler condition @!lm=@ij = @!jm=@il at i = 0 furnishes the equality a 0 jklm = a 0 lkjm ; in particular a 0 jklm is invariant under all permutations of j; k; l. If one puts zm = im + 1 3 X j;k;l a 0 jklm ij ikil; 1 ^ m ^ n; then from (5.13) one finds dzm = dim + X j;k;l a 0 jklm ijikdil; 1 ^ m ^ n; ! = i X 1^m^n dzm ^ dzm + i X j;k;l;m ajklmiji k dil ^ di m + O(jij 3 ); ! = i X 1^m^n dzm ^ dzm + i X j;k;l;m ajklmzjzkdzl ^ dzm + O(jzj 3 ): (5.14) It is now easy to calculate the Chern curvature tensor \Theta (TX )x 0 in terms of the coefficients ajklm and to verify that cjklm = \Gamma ajklm. We leave this as an exercise for the reader. 6. Fundamental identities of K"ahlerian geometry 6.A. Hermitian geometric operators. Assume given (X; !) a Hermitian manifold and let zj = xj + iyj; 1 ^ j ^ n, be C -analytic coordinates about a point a 2 X, such that !(a) = i P dzj ^ dzj is diagonalized at this point. The associated Hermitian form is h(a) = 2 P dzj \Omega dzj and its real part is the Euclidean metric 2 P (dxj ) 2 + (dyj) 2 . It follows that jdxj j = jdyjj = 1= p 2; jdzjj = jdzjj = 1, and that (@=@z1; : : : ; @=@zn) is an orthonormal basis of (T \Lambda a X; !). Formula (4.1) for uj; vk in the orthogonal sum (C \Omega TX) \Lambda = T \Lambda X \Phi T \Lambda X defines a natural inner product on the exterior algebra \Lambda ffl (C \Omega TX) \Lambda . The norm of a form u = X I;J uI;JdzI ^ dzJ 2 \Lambda ffl (C \Omega TX) \Lambda : at a point a is then given by (6.1) ju(a)j 2 = X I;J juI;J (a)j 2 : The Hodge ? operator (4.2) can be extended to the complex-valued forms by the formula (6.2) u ^ ?v = hu; vidV: It follows that ? is a C -linear isometry ? : \Lambda p;q T \Lambda X ! \Lambda n\Gamma q;n\Gamma p T \Lambda X : 25 6. fundamental identities of k"ahlerian geometry The standard Hermitian geometric operators are the operators d; ffi = \Gamma ? d?, the Laplacian \Delta = dffi + ffid already defined, and their complex analogues (6.3) 8 ? ! ? : d = d 0 + d 00 ; ffi = d 0\Lambda + d 00\Lambda ; d 0\Lambda = (d 0 ) \Lambda = \Gamma ? d 00 ?; d 00\Lambda = (d 00 ) \Lambda = \Gamma ? d 0 ?; \Delta 0 = d 0 d 0\Lambda + d 0\Lambda d 0 ; \Delta 00 = d 00 d 00\Lambda + d 00\Lambda d 00 : We say that an operator is of pure degree r if it transforms a form of degree k to a form of degree k + r, and similarly an operator of pure bidegree (s; t) is an operator which transforms the (p; q)-forms to forms of bidegree, (p + s; q + t). (Its total degree is then of course r = s + t.) Thus d 0 ; d 00 ; d 0\Lambda ; d 00\Lambda ; \Delta 0 ; \Delta 00 are of bidegree (1; 0); (0; 1); (\Gamma 1; 0); (0; \Gamma 1); (0; 0); (0; 0) respectively. Another important operator is the operator L of bidegree (1; 1) defined by (6.4) Lu = ! ^ u; and its adjoint \Lambda = L \Lambda = ? \Gamma 1 L? of bidegree (\Gamma 1; \Gamma 1): (6.5) hu; \Lambda vi = hLu; vi: We observe that the unitary group U(TX) ' U(n) has a natural action on the space of (p; q)-forms, given by U(n) \Theta \Lambda p;q T \Lambda X 3 (g; v) 7! (g \Gamma 1 ) \Lambda v: This action makes \Lambda p;q T \Lambda X a unitary representation of U(n). Since the metric ! is invariant, it is clear that L and \Lambda commute with the action of U(n). 6.B. Commutivity identities. If A; B are endomorphisms (of pure degree) of the graded module M ffl = C 1 (X; \Lambda ffl;ffl T \Lambda X ), their graded commutator (or graded Lie bracket) is defined by (6.6) [A; B] = AB \Gamma (\Gamma 1) ab BA where a; b are the degrees of A and B respectively. If C is another endomorphism of degree c, one has the following formal Jacobi identity. (6.7) (\Gamma 1) ca \Theta A; [B; C] \Lambda + (\Gamma 1) ab \Theta B; [C; A] \Lambda + (\Gamma 1) bc \Theta C; [A; B] \Lambda = 0: For all ff 2 \Lambda p;q T \Lambda X , we will still denote by ff the associated endomorphism of type (p; q), operating on \Lambda ffl;ffl T \Lambda X by the formula u 7! ff ^ u. Let fl 2 \Lambda 1;1 T \Lambda X be a real (1; 1)-form. There exists a !-orthogonal basis (i1; i2; : : : ; in) of TX which diagonalizes the two forms ! and fl simultaneously: ! = i X 1^j^n i \Lambda j ^ i \Lambda j ; fl = i X 1^j^n flji \Lambda j ^ i \Lambda j ; flj 2 R: 6.8. Proposition. For any form u = P uj;ki \Lambda J ^ i \Lambda K , one has [fl; \Lambda ]u = X J;K ` X j2J flj + X j2K flj \Gamma X 1^j^n flj ' uJ;Ki \Lambda J ^ i \Lambda K : j.-p. demailly, part i: L 2 hodge theory 26 Proof. If u is of type (p; q), a brutal calculation gives \Lambda u = i(\Gamma 1) p X J;K;l uJ;K(ilyi \Lambda J ) ^ (i l yi \Lambda K ); 1 ^ l ^ n; fl ^ u = i(\Gamma 1) p X J;K;m flmuJ;Ki \Lambda m ^ i \Lambda J ^ i \Lambda m ^ i \Lambda K ; 1 ^ m ^ n; [fl; \Lambda ]u = X J;K;l;m flmuJ;K ` \Gamma i \Lambda l ^ (imyi \Lambda J ) \Delta ^ \Gamma i \Lambda l ^ (i m yi \Lambda K ) \Delta \Gamma \Gamma imy(i \Lambda l ^ i \Lambda J ) \Delta ^ \Gamma i m y(i \Lambda l ^ i \Lambda K ) \Delta ' = X J;K;m flmuJ;K ` i \Lambda m ^ (imyi \Lambda J ) ^ i \Lambda K + i \Lambda J ^ i \Lambda m ^ (i m yi \Lambda K ) \Gamma i \Lambda J ^ i \Lambda K ' = X J;K ` X m2J flm + X m2K flm \Gamma X 1^m^n flm ' uJ;Ki \Lambda J ^ i \Lambda K : \Lambda 6.9. Corollary. For all u 2 \Lambda p;q T \Lambda X , one has [L; \Lambda ]u = (p + q \Gamma n)u. Proof. Indeed, if fl = !, the eigenvalues of fl are fl1 = \Delta \Delta \Delta = fln = 1. \Lambda We introduce the operator B = [L; \Lambda ] which satisfies Bu = (p + q \Gamma n)u for u of bidegree (p; q). Since L has degree 2, one immediately obtains [B; L] = 2L, and similarly [B; \Lambda ] = \Gamma 2\Lambda . This suggests introducing the Lie algebra sl(2; C ) (matrices with zero trace, with the usual commutator bracket [ff; fi] = fffi \Gamma fiff of matrices), for which the basis of 3 matrices (6.10) ` = ` 0 0 1 0 ' ; * = ` 0 1 0 0 ' ; b = ` \Gamma 1 0 0 1 ' satisfies the commutivity relations [`; *] = b; [b; `] = 2`; [b; *] = \Gamma 2*: 6.11. Corollary. There is a natural action of the Lie algebra sl(2; C ) on the vector space \Lambda ffl;ffl T \Lambda X , i.e. a morphism of Lie algebras ae : sl(2; C ) ! End(\Lambda ffl;ffl T \Lambda X ), given by ae(`) = L, ae(*) = \Lambda , ae(b) = B. We now mention the other very important commutivity identities. Let us first assume that X = \Omega ae C n is open in C n and that ! is the standard K"ahler metric, ! = i X 1^j^n dzj ^ dzj: For any form u 2 C 1 (\Omega ; \Lambda p;q T \Lambda X ) one has d 0 u = X I;J;k @uI;J @zk dzk ^ dzI ^ dzJ ;(6.12 0 ) d 00 u = X I;J;k @uI;J @zk dzk ^ dzI ^ dzJ :(6.12 00 ) 27 6. fundamental identities of k"ahlerian geometry Since the global L 2 scalar product is given by hhu; vii = Z \Omega X I;J uI;JvI;J dV; some elementary calculations similar to those of the example in 4.12 show that d 0\Lambda u = \Gamma X I;J;k @uI;J @zk @ @zk y(dzI ^ dzJ );(6.13 0 ) d 00\Lambda = \Gamma X I;J;k @uI;J @zk @ @zk y(dzi ^ dzJ ):(6.13 00 ) We first state a lemma due to Akizuki and Nakano [AN54]. 6.14. Lemma. In C n , one has [d 00\Lambda ; L] = id 0 . Proof. Formula (6.13 00 ) can more succinctly be written d 00\Lambda u = \Gamma X k @ @zk y ` @u @zk ' : We then obtain [d 00\Lambda ; L]u = \Gamma X k @ @zk y ` @ @zk (! ^ u) ' + ! ^ X k @ @zk y ` @u @zk ' : Since ! has constant coefficients, one has @ @zk (! ^ u) = ! ^ @u @zk and consequently [d 00\Lambda ; L]u = \Gamma X k ` @ @zk y \Gamma ! ^ @u @zk \Delta \Gamma ! ^ \Gamma @ @zk y @u @zk \Delta ' = \Gamma X k ` @ @zk y ! ' ^ @u @zk : However, it is clear that @ @zk y! = idzk, therefore [d 00\Lambda ; L]u = i X k dzk ^ @u @zk = id 0 u: \Lambda We are now ready to establish the basic commutivity relations in the situation of an arbitrary K"ahler manifold (X; !). 6.15. Theorem. If (X; !) is K"ahler, then [d 00\Lambda ; L] = id 0 ; [d 0\Lambda ; L] = \Gamma id 00 ; [\Lambda ; d 00 ] = \Gamma id 0\Lambda ; [\Lambda ; d 0 ] = id 00\Lambda : j.-p. demailly, part i: L 2 hodge theory 28 Proof. It suffices to establish the first relation, since the second is the conjugate of the first, and the relations in the second line are the adjoint of the relations in the first line. If (zj) is a geodesic coordinate system at a point x0 2 X, then for all (p; q)-forms u; v with compact support in a neighbourhood of x0, (5.9) implies that hhu; vii = Z M ` X IJ uIJ vIJ + X I;J;K;L aIJKLuIJ vKL ' dV; with aIJKL(z) = O(jzj 2 ) at x0. An integration by parts analogous to that used to obtain (4.12) and (6.13 00 ) gives d 00\Lambda u = \Gamma X I;J;k @uI;J @zk @ @zk y(dzI ^ dzJ ) + X I;J;K;L bIJKLuIJ dzk ^ dzL; where the coefficients bIJKL are obtained by differentiation of aIJKL. Consequently we have bIJKL = O(jzj). Since @!=@zk = O(jzj), the proof of lemma 6.14 above implies [d 00\Lambda ; L]u = id 0 u + O(jzj). In particular the two terms coincide at the given point x0 2 X. \Lambda 6.16. Corollary. If (X; !) is K"ahler, the complex Laplace-Beltrami operators satisfy \Delta 0 = \Delta 00 = 1 2 \Delta : Proof. We first show that \Delta 00 = \Delta 0 . One has \Delta 00 = [d 00 ; d 00\Lambda ] = \Gamma i \Theta d 00 ; [\Lambda ; d 0 ] \Lambda : Since [d 0 ; d 00 ] = 0, the Jacobi identity (6.7) implies that \Gamma \Theta d 00 ; [\Lambda ; d 0 ] \Lambda + \Theta d 0 ; [d 00 ; \Lambda ] \Lambda = 0; hence \Delta 00 = \Theta d 0 ; \Gamma i[d 00 ; \Lambda ] \Lambda = [d 0 ; d 0\Lambda ] = \Delta 0 . Furthermore, \Delta = [d 0 + d 00 ; d 0\Lambda + d 00\Lambda ] = \Delta 0 + \Delta 00 + [d 0 ; d 00\Lambda ] + [d 00 ; d 0\Lambda ]: It therefore suffices to prove: 6.17. Lemma. [d 0 ; d 00\Lambda ] = 0; [d 00 ; d 0\Lambda ] = 0. Proof. We have [d 0 ; d 00\Lambda ] = \Gamma i \Theta d 0 ; [\Lambda ; d 0 ] \Lambda and (6.7) implies that \Gamma \Theta d 0 ; [\Lambda ; d 0 ] \Lambda + \Theta \Lambda ; [d 0 ; d 0 ] \Lambda + \Theta d 0 ; [d 0 ; \Lambda ] \Lambda = 0; hence \Gamma 2 \Theta d 0 ; [\Lambda ; d 0 ] \Lambda = 0 and [d 0 ; d 00\Lambda ] = 0. The second relation [d 00 ; d 0\Lambda ] = 0 is the adjoint of the first. \Lambda 6.18. Theorem. If (X; !) is K"ahler, \Delta commutes with all the operators ?; d 0 ; d 00 ; d 0\Lambda ; d 00\Lambda ; L; \Lambda . Proof. The identities [d 0 ; \Delta 0 ] = [d 0\Lambda ; \Delta 0 ] = 0; [d 00 ; \Delta 00 ] = [d 00\Lambda ; \Delta 00 ] = 0 and [\Delta ; ?] = 0 are immediate. Moreover, the equality [d 0 ; L] = d 0 ! = 0, combined with the Jacobi identity, implies that [L; \Delta 0 ] = \Theta L; [d 0 ; d 0\Lambda ] \Lambda = \Gamma \Theta d 0 ; [d 0\Lambda ; L] \Lambda = i[d 0 ; d 00 ] = 0: Taking adjoints, we obtain [\Delta 0 ; \Lambda ] = 0. \Lambda 29 6. fundamental identities of k"ahlerian geometry 6.C. Primitive elements and the Lefschetz isomorphism theorem. To establish the Lefschetz Theorem, it is convenient to use the representation of sl(2; C ) exhibited in Cor. 6.11. We first recall that if g is a Lie sub-algebra (real or complex) of the Lie algebra sl(r; C ) = End(C r ) of complex matrices and if G = exp(g) ae GL(r; C ) is the associated Lie group, a representation ae : g ! End(V ) of the Lie algebra in a complex vector space V induces by exponentiation a representation ~ae : G ! GL(V ) of the group G. Conversely, a representation ~ae : G ! GL(V ) induces by differentiation a representation ae : g ! End(V ) of Lie algebras; there is therefore an identification between these two notions. If G is compact, a classical lemma of H. Weyl shows that all representations of g are broken down into a direct sum of irreducible representations (one says that g is reductive): the Haar measure of G indeed allows the construction of an invariant Hermitian metric on V , and one exploits the fact that the orthogonal complement of a sub-representation is a sub-representation. In particular the Lie algebra su(r) of the compact group SU(r) is reductive. It is the same as for sl(r; C ), which is the complexification of su(r). We will need the following well-known lemma from representation theory. 6.19. Lemma. Let ae : sl(2; C ) ! End(V ) be a representation of the Lie algebra sl(2; C ) on a finite dimensional complex vector space V , and let L = ae(`); \Lambda = ae(*); B = ae(b) 2 End(V ) be the endomorphisms of V associated to the basis elements of sl(2; C ). Then: a) V = \Phi _2ZV_ is a (finite) direct sum of eigenspaces of B, whose eigenvalues _ are integers. An element v 2 V_ is said to be an element of pure weight _. b) L and \Lambda are nilpotent, satisfying L(V_) ae V_+2, \Lambda (V_) ae V_\Gamma 2 for all _ 2 Z. c) We denote by P = Ker \Lambda = fv 2 V ; \Lambda v = 0g, the set of primitive elements. One then has a direct sum decomposition V = M r2N L r (P ): d) V is isomorphic to a finite direct sum \Phi m2NS(m) \Phi ffm of irreducible representations, where S(m) ' S m (C 2 ) is the representation of sl(2; C ) induced by the m-th symmetric product of the natural representation of SL(2; C ) on C 2 , and ffm = dim Pm is the multiplicity of the isotypic component S(m). e) If P_ = P " V_, then P_ = 0 for _ ? 0 and P = \Phi _2Z;_^0P_. The endomorphism L r : P\Gamma m ! Vm+2r is injective for r ^ m and zero for r ? m. f) V_ = L r2N;r*_ L r (P_\Gamma 2r), where L r : P_\Gamma 2r ! L r (P_\Gamma 2r) is bijective. g) For any r 2 N, the endomorphism L r : V\Gamma r ! Vr is bijective. Proof. We first observe the following fact: If v 2 V_, then Lv has pure weight _ + 2 and \Lambda v has pure weight _ \Gamma 2. Indeed, one has BLv = LBv + [B; L]v = L(_v) + 2Lv = (_ + 2)Lv; B\Lambda v = \Lambda Bv + [B; \Lambda ]v = \Lambda (_v) \Gamma 2\Lambda v = (_ \Gamma 2)\Lambda v: Now suppose V 6= 0 and let v 2 V_ be a non-zero eigenvector. If the vectors (\Lambda k v)k2N were all non-zero, one would have an infinite number of eigenvectors of B with _ \Gamma 2k distinct eigenvalues, which is impossible. Therefore there exists an integer r * 0 such that \Lambda r v 6= 0 and \Lambda k v = 0 for k ? r. Consequently \Lambda r v is a non-zero primitive element of pure weight _ 0 = _ \Gamma 2r. Thus we conclude that for j.-p. demailly, part i: L 2 hodge theory 30 some _ 2 C , there exists w 2 P , a non-zero element of pure weight _. The same reasoning as above applied to the powers L k w shows that there exists an integer m ? 0 such that L m w 6= 0 and L m+1 w = 0. The vector space W of dimension m + 1 generated by wk = L k w; 0 ^ k ^ m is stable under the action of sl(2; C ). Indeed one has Bwk = (_ + 2k)wk; Lwk = wk+1 by definition, while \Lambda wk = \Lambda L k w = L k \Lambda w \Gamma X 0^j^k\Gamma 1 L k\Gamma j\Gamma 1 [L; \Lambda ]L j w = 0 \Gamma X L k\Gamma j\Gamma 1 BL j w = \Gamma X 0^j^k\Gamma 1 (_ + 2j)L k\Gamma 1 w = k(\Gamma _ \Gamma k + 1)wk\Gamma 1: By applying this relation to the indice k = m + 1 for which wm+1 = 0, it follows that one must necessarily have _ = \Gamma m ^ 0. We remark that B_W is diagonalizable (the eigenvectors of W being the vectors wk of integral weight 2k \Gamma m), and that the primitive elements of W are reduced to the line C w, such that W = \Phi L r (C w). Properties (a,b,c,d) mentioned above are then easily obtained by induction on dim V . By considering the quotient representation V=W one can argue by induction that the eigenvalues of B are integers and that L; \Lambda are nilpotent. It is easy to verify that W ' S m (C 2 ) as a representation of SL(2; C ). (If e1; e2 are two basis vectors of C 2 , the isomorphism sends w = w0 to e m 1 and wk to L k e m 1 = m(m \Gamma 1) \Delta \Delta \Delta (m \Gamma k + 1)e k 1 e m\Gamma k 2 .) The fact that one has a direct sum of representations V = V 0 \Phi W (with V 0 = W ? ae V for a certain SU(2; C )-invariant metric) involves the diagonalizability of B, by induction on dim V , as well as the formula V = \Phi L r (P ) and the decomposition in d). e) The relation [B; \Lambda ] = \Gamma 2\Lambda shows that P = Ker \Lambda is stable under B, consequently P = M (P " V_) = M P_: The above calculations show that the non-zero primitive elements w are of weight \Gamma m ^ 0, so that P_ = 0 if _ ? 0. The latter assertion of e) follows from the fact that for 0 6= w 2 P\Gamma m, one has L r w 6= 0 if and only if r ^ m. f) An immediate consequence of e) and the decomposition V = \Phi r2NL r (P ), if one restricts only to elements of pure weight _. One can only have L r (P_\Gamma 2r) 6= 0 if either r ^ m = \Gamma (_ \Gamma 2r), or r * _. g) It suffices to verify the assertion in the case of an irreducible representation V ' S m (C 2 ). In this case, the result is clear, since the weights 2k \Gamma m; 0 ^ k ^ m are distributed symmetrically in the interval [\Gamma m; m] and that V is generated by (L k w)0^k^m for any non-zero vector w of V\Gamma m. \Lambda We now interpret these results in the case of a representation of sl(2; C ) on V = \Lambda ffl;ffl T \Lambda X . The component \Lambda k (C \Omega TX ) \Lambda = \Phi p+q=k\Lambda p;q T \Lambda X can then be identified with the eigenspace V_ of B of weight _ = k \Gamma n = p + q \Gamma n (by definition of B, see (6.9)). 6.20. Definition. A homogeneous form u 2 \Lambda k (C k \Omega TX) \Lambda is called primitive if \Lambda u = 0. The space of primitive forms of total degree k is denoted by Prim k T \Lambda X = M p+q=k Prim p;q T \Lambda X : 31 6. fundamental identities of k"ahlerian geometry Since the operator \Lambda commutes with the action of U(TX ) ' U(n) on the exterior algebra, it is clear that Prim p;q T \Lambda X ae \Lambda p;q T \Lambda X is a U(n)-invariant subspace. One further sees (prop. 6.24) that Prim p;q T \Lambda X is in fact an irreducible representation of U(n). Properties (6.19 e, f, g) successively imply 6.21. Proposition. We have Prim k T \Lambda X = 0 for k ? n. Moreover, if u 2 Prim k T \Lambda X , k ^ n, then L r u = 0 for r ? n \Gamma k. 6.22. Primitive decomposition formula. For any u 2 \Lambda k (C \Omega TX ) \Lambda , there exists a unique decomposition u = X r*(k\Gamma n)+ L r uk\Gamma 2r; uk\Gamma 2r 2 Prim k\Gamma 2r T \Lambda X : Consequently, one obtains a decomposition into a direct sum of representations of U(n) \Lambda k (C \Omega TX) \Lambda = M r*(k\Gamma n)+ L r Prim k\Gamma 2r T \Lambda X ; \Lambda p;q (C \Omega TX) \Lambda = M r*(p+q\Gamma n)+ L r Prim p\Gamma r;q\Gamma r T \Lambda X : 6.23. Lefschetz Isomorphism Theorem. The linear operators L n\Gamma k : \Lambda k (C \Omega TX ) \Lambda ! \Lambda 2n\Gamma k (C \Omega TX ) \Lambda ; L n\Gamma p\Gamma q : \Lambda p;q T \Lambda X ! \Lambda n\Gamma q;n\Gamma p T \Lambda X ; are isomorphisms for all integers k ^ n and (p; q) satisfying p + q ^ n. 6.24. Proposition. For any (p; q) 2 N 2 satisfying p + q ^ n, Prim p;q T \Lambda X is an irreducible representation of U(n); more precisely, it is the irreducible representation associated to the highest weight ffl1 + \Delta \Delta \Delta + fflq \Gamma (ffln\Gamma p+1 + \Delta \Delta \Delta + ffln), where (fflj) is the canonical basis of characters of the maximal commutative subgroup U(1) n ae U(n). The primitive decomposition of \Lambda p;q T \Lambda X or of \Lambda k (C \Omega TX ) \Lambda is the same as the decomposition into irreducible components under the action of U(n). Proof. First observe that Prim p;q T \Lambda X 6= 0, since for example dz1 ^ \Delta \Delta \Delta ^ dzp ^ dzp+1 ^ \Delta \Delta \Delta ^ dzp+q 2 Prim p;q T \Lambda X : Further, the primitive decomposition gives \Lambda p;q T \Lambda X = M 0^r^m L r Prim p\Gamma r;q\Gamma r T \Lambda X with m = min(p; q), which shows that the U(n)-module \Lambda p;q T \Lambda X has at least m + 1 non-trivial irreducible components, to account for each of the terms Prim p\Gamma r;q\Gamma r T \Lambda X , 0 ^ r ^ m. To see that these are irreducible , it suffices to show that the U(n)- module \Lambda p;q T \Lambda X has no more than (m + 1) irreducible components. However, by complexification of the representation of U(n), one obtains a representation isomorphic to that of GL(n; C ) on \Lambda p T \Lambda X \Omega \Lambda q TX given by g \Delta (u \Omega ,) = (g \Gamma 1 ) \Lambda u \Omega g\Lambda ,. The representation theory of linear groups shows that the irreducible components of a representation are in bijective correspondence with the eigenvectors associated to j.-p. demailly, part i: L 2 hodge theory 32 the action of the Borel subgroup Bn of upper triangular matrices. We leave it to the reader to show that these eigenvectors correspond precisely to the (p; q)-forms L r (dzn\Gamma p+r+1 ^ \Delta \Delta \Delta ^ dzn ^ dz1 ^ \Delta \Delta \Delta ^ dzq\Gamma r); 0 ^ r ^ m; for which the weight under the action of U(1) n is ffl1 + \Delta \Delta \Delta + fflq\Gamma r \Gamma (ffln\Gamma p+r+1 + \Delta \Delta \Delta + ffln). \Lambda 7. The groups H p;q (X; E)H p;q (X; E)H p;q (X; E) and Serre duality We now arrive at some holomorphic consequences of Hodge theory. A large part of this theory was developed by K. Kodaira, S. Lefschetz and A. Weil. The reader can profitably consult the Completed Works of Kodaira [Kod75] and the book by A. Weil [Wei57]; see also [Wel80] for a more recent account. Let (X; !) be a compact Hermitian manifold and E a Hermitian holomorphic vector bundle of rank r over X. We will denote by DE the Chern connection of E, D \Lambda E = \Gamma ? DE? the formal adjoint of DE, and D 0\Lambda E , D 00\Lambda E the components of D \Lambda E of type (\Gamma 1; 0) and (0; \Gamma 1). A similar calculation to that done in 4.14 shows that oeD00 E (x; ,) \Delta s = , 0;1 ^ s; , 2 R T \Lambda X = HomR(TX ; R); s 2 Ex; where , (0;1) is the type (0; 1) part of the real 1-form ,. Consequently, we see that the principal part of the operator \Delta 00 E = D 00 E D 00\Lambda E + D 00\Lambda E D 00 E is given by oe\Delta 00 E (x; ,) \Delta s = \Gamma j, 0;1 j 2 s = \Gamma 1 2 j,j 2 s; and there is a similar result for \Delta 0 E . In particular oe\Delta 0 E = oe\Delta 00 E = 1 2 oe\Delta E and \Delta 00 E is a self-adjoint elliptic operator on each of the spaces C 1 (X; \Lambda p;q T \Lambda X \Omega E). Using D 002 E = 0, one arrives at the following result, in the same way as obtained in x4.C. 7.1. Theorem. For any bidegree (p; q), there exists an orthogonal decomposition C 1 (X; \Lambda p;q T \Lambda X \Omega E) = H p;q (X; E) \Phi Im D 00 E \Phi Im D 00\Lambda E where H p;q (X; E) is the space of \Delta 00 E -harmonic forms in C 1 (X; \Lambda p;q T \Lambda X \Omega E). The above decomposition shows that the subspace of q-cocycles of the complex (C 1 (X; \Lambda p;ffl T \Lambda X \Omega E); d 00 ) is H p;q (X; E) \Phi Im D 00 E . From here, we deduce the 7.2. Theorem (Hodge isomorphism). The Dolbeault cohomology groups H p;q (X; E) are finite dimensional, and there is an isomorphism H p;q (X; E) ' H p;q (X; E): Another interesting consequence is a proof of the Serre duality theorem for compact complex manifolds. See Serre [Ser55] for a proof in a somewhat more general context. 7.3. Theorem (Serre duality). The bilinear pairing H p;q (X; E) \Theta H n\Gamma p;n\Gamma q (X; E \Lambda ) ! C ; (s; t) 7! Z M s ^ t is a non-degenerate duality. 33 8. comohology of compact k"ahler manifolds Proof. Let s1 2 C 1 (X; \Lambda p;q T \Lambda X \Omega E); s2 2 C 1 (X; \Lambda n\Gamma p;n\Gamma q\Gamma 1 T \Lambda X \Omega E). Since s1 ^ s2 is of bidegree (n; n \Gamma 1), we have (7.4) d(s1 ^ s2) = d 00 (s1 ^ s2) = d 00 s1 ^ s2 + (\Gamma 1) p+q s1 ^ d 00 s2: Stokes theorem implies that the bilinear pairing above can be factored through the Dolbeault cohomology groups. The operator # defined is x4.A satisfies # : C 1 (X; \Lambda p;q T \Lambda X \Omega E) ! C 1 (X; \Lambda n\Gamma p;n\Gamma q T \Lambda X \Omega E \Lambda ): Moreover, (4.20) implies D 00 E\Lambda (#s) = (\Gamma 1) deg s #(D 00 E ) \Lambda s; (D 00 E\Lambda ) \Lambda (#s) = (\Gamma 1) deg s+1 #D 00\Lambda E s; \Delta 00 E\Lambda (#s) = #\Delta 00 E s; where DE\Lambda is the Chern connection of E \Lambda . Consequently, s 2 H p;q (X; E) if and only if #s 2 H n\Gamma p;n\Gamma q (X; E \Lambda ). Theorem 7.3 is then a consequence of the fact that the integral jjsjj 2 = R X s ^ #s is non-vanishing if s 6= 0. 8. Comohology of compact K"ahler manifolds 8.A. Bott-Chern cohomology groups. Let X be a complex manifold, for the moment not necessarily compact. The following "cohomology groups" are useful for describing certain aspects of the Hodge theory of compact complex manifolds, which are not necessarily K"ahler. 8.1. Definition. The Bott-Chern cohomology groups of X are given by H p;q BC (X; C ) = (C 1 (X; \Lambda p;q T \Lambda X ) " Ker d)=d 0 d 00 C 1 (X; \Lambda p\Gamma 1;q\Gamma 1 T \Lambda X ): The cohomology H ffl;ffl BC (X; C ) has a bigraded algebra structure, which we call the Bott-Chern cohomology algebra of X. Since the group d 0 d 00 C 1 (X; \Lambda p\Gamma 1;q\Gamma 1 T \Lambda X ) is also contained in the group of coboundaries d 00 C 1 (X; \Lambda p;q\Gamma 1 T \Lambda X ) of the Dolbeault complex as well as that in coboundaries of the de Rham complex dC 1 (X; \Lambda p+q\Gamma 1 (C \Omega TX ) \Lambda ), there are canonical morphisms H p;q BC (X; C ) ! H p;q (X; C );(8.2) H p;q BC (X; C ) ! H p+q (X; C );(8.3) of the Bott-Chern cohomology to the Dolbeault or de Rham cohomology. These morphisms are C -algebra homomorphisms. It is also clear from the definition that we have the symmetry property H q;p BC (X; C ) = H p;q BC (X; C ). One can show from the Hodge-Fr"olicher spectral sequence (see x10) that H p;q BC (X; C ) is always finite dimensional if X is compact. 8.B. Hodge decomposition theorem. We assume from now on that (X; !) is a compact K"ahler manifold. The equality \Delta = 2\Delta 00 shows that \Delta is homogeneous with respect to bidegree and that there is an orthogonal decomposition (8.4) H k (X; C ) = M p+q=k H p;q (X; C ): j.-p. demailly, part i: L 2 hodge theory 34 Since \Delta 00 = \Delta 0 = \Delta 00 , one has the equality H q;p (X; C ) = H p;q (X; C ). By applying the Hodge isomorphism theorem for de Rham cohomology and for Dolbeault cohomology, one obtains: 8.5. Theorem (Hodge Decomposition). On a compact K"ahler manifold, there are canonical isomorphisms H k DR (X; C ) ' M p+q=k H p;q (X; C ) (Hodge decomposition), H q;p (X; C ) ' H p;q (X; C ) (Hodge symmetry). The only point that is not a priori obvious is that isomorphisms are independent of the choice of K"ahler metric. To show that this is indeed the case, one can use the following lemma, which will allow us to compare the three types of cohomology groups considered in x8.A. 8.6. Lemma. Let u be a d-closed (p; q)-form. The following properties are equivalent: a) u is d-exact; b 0 ) u is d 0 -exact; b 00 ) u is d 00 -exact; c) u is d 0 d 00 -exact, i.e. u can be written u = d 0 d 00 v. d) u is orthogonal to H p;q (X; C ). Proof. It is evident that c) implies a), b 0 ), b 00 ), and that a) or b 0 ) or b 00 ) implies d). It suffices therefore to prove that d) implies c). Since du = 0, we have d 0 u = d 00 u = 0, and since u is assumed orthogonal to H p;q (X; C ), th. 7.1 implies that u = d 00 s, s 2 C 1 (X; \Lambda p;q\Gamma 1 T \Lambda X ). The analogous theorem to th. 7.1 for d 0 (which can be deduced by complex conjugation) shows that one can write s = h + d 0 v + d 0\Lambda w, where h 2 H p;q\Gamma 1 (X; C ), v 2 C 1 (X; \Lambda p\Gamma 1;q\Gamma 1 T \Lambda X ) and w 2 C 1 (X; \Lambda p+1;q\Gamma 1 T \Lambda X ). Consequently u = d 00 d 0 v + d 00 d 0\Lambda w = \Gamma d 0 d 00 v \Gamma d 0\Lambda d 00 w by an application of Lemma 6.16. Since d 0 u = 0, the component d 0\Lambda d 00 w orthogonal to Ker d 0 must be zero. \Lambda From Lemma 8.6 we deduce the following corollary, which in turn implies that the Hodge decomposition does not depend on the choice of K"ahler metric. 8.7. Corollary. Let X be a compact K"ahler manifold. Then the natural morphisms H p;q BC (X; C ) ! H p;q (X; C ); M p+q=k H p;q BC (X; C ) ! H k DR (X; C ) are isomorphisms. Proof. The surjectivity of H p;q BC (X; C ) ! H p;q (X; C ) follows from the fact that any class in H p;q (X; C ) can be represented by a harmonic (p; q)-form, therefore by a d-closed (p; q)-form; the injectivity property is nothing more than the equivalence (8.5b 00 ) , (8.5c). Therefore H p;q BC (X; C ) ' H p;q (X; C ) ' H p;q (X; C ), and the isomorphism M p+q=k H p;q BC (X; C ) ! H k DR (X; C ) 35 8. comohology of compact k"ahler manifolds is a consequence of (8.4). \Lambda We now mention two simple consequences of Hodge theory. The first concerns the calculation of the Dolbeault cohomology of P n . Since H p;p (P n ; C ) contains the non-zero class f! p g and since H 2p DR (P n ; C ) = C , the Hodge decomposition formula implies: 8.8. Consequence. The Dolbeault cohomology groups of P n are H p;p (P n ; C ) = C for 0 ^ p ^ n; H p;q (P n ; C ) = 0 for p 6= q: \Lambda 8.9. Proposition. Any holomorphic p-form on a compact K"ahler manifold X is d-closed. Proof. If u is a holomorphic form of type (p; 0) then d 00 u = 0. Moreover d 00\Lambda u is of type (p; \Gamma 1), hence d 00\Lambda u = 0. Consequently \Delta u = 2\Delta 00 u = 0, which implies that du = 0. \Lambda 8.10. Example. Consider the Heisenberg group G ae Gl3(C ), defined by the subgroup of matrices M = 0 @ 1 x z 0 1 y 0 0 1 1 A ; (x; y; z) 2 C 3 : Let \Gamma be a discrete subgroup of matrices with the property that the coefficients x; y; z belong to the ring Z[i] (or more generally in the ring of imaginary quadratic integers). Then X = G=\Gamma is a compact complex manifold of dimension 3, called a Iwasawa manifold. The equality M \Gamma 1 dM = 0 @ 0 dx dz \Gamma xdy 0 0 dy 0 0 0 1 A ; shows that dx; dy; dz \Gamma xdy are left invariant 1-forms on G. These forms induce holomorphic 1-forms on the quotient X = G=\Gamma . Since dz \Gamma xdy is not d-closed, one concludes that X cannot be K"ahler. 8.11. Remark. For simplicity of notation we work here with constant coefficients, but the reader can easily verify that one has analogous results for cohomology with values in a local system of coefficients (flat Hermitian bundle), as in x4.C. It is enough to replace everywhere in the proof the operator d = d 0 + d 00 by DE = D 0 E +D 00 E , and to observe that one still has \Delta 0 E = \Delta 00 E = 1 2 \Delta E (proof identical to that of Cor. (6.16)). One can then deduce the existence of isomorphisms H p;q BC (X; E) ! H p;q (X; E); M p+q=k H p;q BC (X; E) ! H k DR (X; E) and a canonical decomposition H k DR (X; E) = M p+q=k H p;q (X; E): In this context, the symmetry property of Hodge becomes H p;q (X; E) ' H q;p (X; E \Lambda ); j.-p. demailly, part i: L 2 hodge theory 36 via the antilinear operator # considered in x4 and x7. These observations are useful for the study of variations of Hodge structure. 8.C. Primitive decomposition and hard Lefschetz theorems. We first introduce some standard notation. The Betti numbers and the Hodge numbers of X are by definition (8.12) bk = dimC H k (X; C ); h p;q = dimC H p;q (X; C ): According to the Hodge decomposition, the numbers satisfy the relations (8.13) bk = X p+q+k h p;q ; h q;p = h p;q : Consequently, the Betti numbers b2k+1 of a compact K"ahler manifold are even. Note that the Serre duality theorem gives the additional relation h p;q = h n\Gamma p;n\Gamma q , provided that X is compact. As we will see, the existence of the primitive decomposition implies many other interesting characteristic properties of the cohomology algebra of a compact K"ahler manifold. 8.14. Lemma. If u = P r*(k\Gamma n)+ L r ur is the primitive decomposition of a harmonic k-form u, then all the components ur are harmonic. Proof. Since [\Delta ; L] = 0, one obtains 0 = \Delta u = P r L r \Delta ur, therefore \Delta ur = 0 according to the uniqueness of the decomposition. \Lambda Denote by H p prim (X; C ) = L p+q=k H p+q prim (X; C ) the space of primitive harmonic k-forms and let h p;q prim be the dimension of the component of bidegree (p; q). Lemma (8.14) gives H p;q (X; C ) = M r*(p+q\Gamma n)+ L r H p\Gamma r;q\Gamma r prim (X; C );(8.15) h p;q = X r*(p+q\Gamma n)+ h p\Gamma r;q\Gamma r prim :(8.16) Formula (8.16) can be written as (8.16 0 ) 8 ? ! ? : If p + q ^ n; h p;q = h p;q prim + h p\Gamma 1;q\Gamma 1 prim + \Delta \Delta \Delta If p + q * n; h p;q = h n\Gamma q;n\Gamma p prim + h n\Gamma q\Gamma 1;n\Gamma p\Gamma 1 prim + \Delta \Delta \Delta . 8.17. Corollary. The Hodge and Betti numbers satisfy the following inequalities. a) If k = p + q ^ n, then h p;q * h p\Gamma 1;q\Gamma 1 ; bk * bk\Gamma 2, b) If k = p + q * n; then h p;q * h p+1;q+1 ; bk * bk+2. \Lambda Another important result of Hodge theory (that is in fact a direct consequence of Cor. 6.23) is the 8.18. Hard Lefschetz Theorem. The cup product morphisms L n\Gamma k : H k (X; C ) ! H 2n\Gamma k (X; C ); k ^ n; L n\Gamma p\Gamma q : H p;q (X; C ) ! H n\Gamma q;n\Gamma p (X; C ); p + q ^ n; 37 8. comohology of compact k"ahler manifolds are isomorphisms. \Lambda Another way of stating the hard Lefschetz Theorem is to introduce the HodgeRiemann bilinear form on H k DR (X; C ), defined by (8.19) Q(u; v) = (\Gamma 1) k(k\Gamma 1)=2 Z X u ^ v ^ ! n\Gamma k : The hard Lefschetz Theorem combined with Poincar'e duality says that Q is nondegenerate. Moreover Q is of parity (\Gamma 1) k (symmetric if k is even, alternating if k is odd). When ! is a Hodge metric, that is a K"ahler metric such that f!g 2 H 2 (X; Z), it is clear that Q takes integer values when restricted to H k (X; Z)=(torsion). The Hodge-Riemann bilinear form satisfies the following additional properties: For p + q = k, Q(H p;q ; H p 0 ;q 0 ) = 0 if (p 0 ; q 0 ) 6= (q; p);(8.20 0 ) If 0 6= u 2 H p;q prim (X; C ); then i p\Gamma q Q(u; u) = jjujj 2 ? 0: (8.20 00 ) In fact (8.20 0 ) is clear and (8.20 00 ) will be shown if we can check that any (p; q)- primitive form u satisfies (\Gamma 1) k(k\Gamma 1)=2 i p\Gamma q ! n\Gamma k ^ u = ?u: Since Prim p;q T \Lambda X is an irreducible representation of U(n), it suffices to verify the formula for a conveniently chosen (p; q)-form u. One can take for example u = dz1 ^ \Delta \Delta \Delta ^ dzp ^ dzp+1 ^ \Delta \Delta \Delta ^ dzp+q from an orthonormal basis for !. The necessary verification is easy for the reader to work out as an exercise. 8.D. A description of the Picard group. Another important application of Hodge theory is a description of the Picard group H 1 (X; O \Lambda ) of a compact K"ahler manifold. We assume here that X is connected. The exponential exact sequence 0 ! Z ! O ! O \Lambda ! 1 gives (8.21) 0 ! H 1 (X; Z) ! H 1 (X; O) ! H 1 (X; O \Lambda ) c1 \Gamma ! H 2 (X; Z) ! H 2 (X; O); taking into account the fact that the map exp(2ssiffl) : H 0 (X; O) = C ! H 0 (X; O \Lambda ) = C \Lambda is surjective. One has H 1 (X; O) ' H 0;1 (X; C ) by the Dolbeault isomorphism theorem. The dimension of this group is called the irregularity of X and it is usually denoted by (8.22) q = q(X) = h 0;1 = h 1;0 : Consequently we have b1 = 2q and (8.23) H 1 (X; O) ' C q ; H 0 (X; \Omega 1 X ) = H 1;0 (X; C ) ' C q : 8.24. Lemma. The image of H 1 (X; Z) in H 1 (X; O) is a lattice. Proof. Consider the morphism H 1 (X; Z) ! H 1 (X; R) ! H 1 (X; C ) ! H 1 (X; O) induced by the inclusions Z ae R ae C ae O. Since the ^ Cech cohomology groups with values in Z or R can be calculated by a finite covering of open sets for which each is diffeomorphic to an open convex set, and the same for all their mutual intersections, it is clear that H 1 (X; Z) is a Z-module of finite type and that the j.-p. demailly, part i: L 2 hodge theory 38 image of the H 1 (X; Z) in H 1 (X; R) is a lattice. It suffices therefore to show that the map H 1 (X; R) ! H 1 (X; O) is an isomorphism. However, the commutative diagram 0 ! C ! A 0 d \Gamma ! A 1 d \Gamma ! A 2 ! \Delta \Delta \Delta # # # # 0 ! O ! A 0;0 d 00 \Gamma ! A 0;1 d 00 \Gamma ! A 0;2 ! \Delta \Delta \Delta shows that the map H 1 (X; R) ! H 1 (X; O) corresponds, for de Rham and Dolbeault cohomology, to the composite map H 1 DR (X; R) ae H 1 DR (X; C ) ! H 0;1 (X; C ): Since H 1;0 (X; C ) and H 0;1 (X; C ) are complex conjugate subspaces in the complexification H 1 DR (X; C ) of H 1 DR (X; R), we can easily deduce that H 1 DR (X; R) ! H 0;1 (X; C ) is an isomorphism. \Lambda As a consequence of this lemma, H 1 (X; Z) is of rank 2q, i.e. H 1 (X; Z) ' Z 2q . The complex torus of dimension q (8.25) Jac(X) = H 1 (X; O)=H 1 (X; Z) is called the Jacobian variety of X. It is isomorphic to the subgroup of H 1 (X; O \Lambda ) corresponding to the line bundles with zero first Chern class. In other words, the kernel of the arrow H 2 (X; Z) ! H 2 (X; O) = H 0;2 (X; C ); which defines the integral cohomology classes of type (1; 1), is equal to the image of the morphism c1(ffl) in H 2 (X; Z). This subgroup is called the N'eron-Severi group of X, and is denoted by N S(X). Its rank ae(X) is called the Picard number of X. The exact sequence (8.21) then gives (8.26) 0 ! Jac(X) ! H 1 (X; O \Lambda ) c1 \Gamma ! N S(X) ! 0: The Picard group H 1 (X; O \Lambda ) is therefore an extension of the complex torus Jac(X) by the Z-module of finite type N S(X). 8.27. Corollary. The Picard group of P n is H 1 (P n ; O \Lambda ) ' Z with O(1) as generator, i.e. any line bundle over P n is isomorphic to one of the line bundles O(k), k 2 Z. Proof. We have H k (P n ; O) = H 0;k (P n ; C ) = 0 for k * 1 by applying conseq. 8.8, therefore Jac(P n ) = 0 and N S(P n ) = H 2 (P n ; Z) ' Z. Moreover, c1(O(1)) is a generator of H 2 (P n ; Z). \Lambda 9. The Hodge-Fr"olicher spectral sequence Assume given X a complex manifold (i.e. not necessarily compact) of dimension n. We consider the double complex K p;q = C 1 (X; \Lambda p;q T \Lambda X ) with its total differential d = d 0 + d 00 . The Hodge-Fr"olicher spectral sequence (or Hodge to de Rham spectral sequence) is by definition the spectral sequence associated to this double complex. 39 9. the hodge-fr"olicher spectral sequence We first recall the algebraic machinery of spectral sequences, which applies to an arbitrary double complex (K p;q ; d 0 +d 00 ) of modules over a ring. We assume here for simplicity that K p;q = 0 if p ! 0 or q ! 0. One first associates to K ffl;ffl the total complex (K ffl ; d) such that K l = \Phi p+q=lK p;q , equipped with the total differential d = d 0 + d 00 . Then K ffl admits a decreasing filtration formed from the subcomplexes F p K ffl where (9.1) F p K l = M p^j^l K j;l\Gamma j : One obtains an induced filtration on the cohomology groups H l (K ffl ) of the total complex by setting (9.2) F p H l (K ffl ) := Im \Gamma H l (F p K ffl ) ! H l (K ffl ) \Delta ; and one denotes by G p H l (K ffl ) = F p H l (K ffl )=F p+1 H l (K ffl ) the associated graded module. The theory of spectral sequences (see for example [God57]) says that there exists a sequence of double complexes E ffl;ffl r ; r * 1, equipped with differentials dr : E p;q r ! E p+r;q\Gamma r+1 r of bidegree (r; \Gamma r + 1) such that Er+1 = H ffl (Er) is calculated recursively as the cohomology of the complex (E ffl;ffl r ; dr), and where the limit E p;q 1 = limr!+1 E p;q r is identified with the graded module G ffl H ffl (K ffl ), more precisely E p;q 1 = G p H p+q (K ffl ). The E1 terms are defined as the cohomology groups of the partial complex d 00 : K p;q ! K p;q+1 by passing to the second differential, that is (9.4) E p;q 1 = H q \Gamma (K p;ffl ; d 00 ) \Delta ; and the differential d1 : E p;q 1 ! E p+1;q 1 is induced by the first differential d 0 : (9.5) d 0 : H q \Gamma (K p;ffl ; d 00 ) \Delta ! H q \Gamma (K p+1;ffl ; d 00 ) \Delta : In fact, one has E p;q r = 0 unless p; q * 0, and the limit E1 = lim Er is stationary, more precisely E p;q r = E p;q r+1 = \Delta \Delta \Delta = E p;q 1 when r * max(p + 1; q + 2); as one sees by considering the indices in which dr can be non-zero. One says that the spectral sequence converges to the graded filtered module H ffl (K ffl ), and it is customary to represent this situation by the notation E p;q 1 ) G p H p+q (K ffl ): A careful examination of the terms of small degree leads to the exact sequence (9.6) 0 ! E 1;0 2 ! H 1 (K ffl ) ! E 0;1 2 d2 \Gamma ! E 2;0 2 ! H 2 (K ffl ): One says that the spectral sequence degenerates at Er 0 if dr = 0 for all r * r0 and for all bidegree (p; q). In this case one has E ffl;ffl r0 = E ffl;ffl r0+1 = \Delta \Delta \Delta = E ffl;ffl 1 . In the case of the Hodge-Fr"olicher spectral sequence, the E1 terms are the Dolbeault cohomology groups E p;q 1 = H p;q (X; C ), and the cohomology of the total complex is precisely the de Rham cohomology H ffl DR (X; C ). One therefore obtains a spectral sequence (9.7) E p;q 1 = H p;q (X; C ) ) G p H p+q DR (X; C ) j.-p. demailly, part i: L 2 hodge theory 40 of the Dolbeault cohomology to the de Rham cohomology. The corresponding filtration F p H k DR (X; C ) of cohomology groups is called the Hodge(-Fr"olicher) filtration. Now assume that X is compact. All the terms E p;q r are then finite dimensional vector spaces. Since Er+1 = H ffl (Er), the dimensions dim E p;q r are decreasing (or stationary) with r, therefore dim E p;q 1 ^ dim E p;q r , and equality takes place if and only if the spectral sequence degenerates at Er. In particular, the Betti numbers bl = dim H l (X; C ) and the Hodge numbers h p;q = dim E p;q 1 satisfy the inequality (9.8) bl = X p+q=l dim E p;q 1 ^ X p+q=l dim E p;q 1 = X p+q=l h p;q ; and equality is equivalent to the degeneration of the spectral sequence at E ffl 1 . As a consequence, we have the 9.9. Theorem. If X is a compact K"ahler manifold, the following properties are equivalent: a) The Hodge-Fr"olicher spectral sequence degenerates at E ffl 1 . b) One has the equality bl = P p+q=l h p;q for all l. c) There exists an isomorphism G p H p+q DR (X; C ) ' H p;q (X; C ) for all p; q. If one of these conditions is satisfied, the isomorphism c) is given in a canonical way. We can now again interpret the results of x8.B as follows. 9.10. Theorem. If X is a compact K"ahler manifold, the Hodge-Fr"olicher spectral sequence degenerates at E1 and there is a canonical decomposition H l DR (X; C ) = M p+q=l H p;q (X; C ); H q;p (X; C ) = H p;q (X; C ): In terms of this decomposition, the filtration F p H l DR (X; C ) is given by F p H l DR (X; C ) = M j*p H j;l\Gamma j (X; C ): In particular, the conjugate filtration F ffl H l DR is opposed to the filtration F ffl H l DR , i.e. H l DR (X; C ) = F p H l DR (X; C ) \Phi F l\Gamma p+1 H l DR (X; C ): 9.11. Definition. If X is a compact complex manifold, we say that X admits a Hodge decomposition if the Hodge-Fr"olicher spectral sequence degenerates at E1 and if the conjugate filtration F ffl H l DR is opposed to F ffl H l DR , i.e. H l DR = F p H l DR \Phi F l\Gamma p+1 H l DR for all p. If X admits a Hodge decomposition in the sense of def. 9.11 and if p + q = l, then it is immediate from the equality H l DR = F p+1 H l DR \Phi F q H l DR that F p H l DR = F p+1 H l DR \Phi (F p H l DR " F q H l DR ): Therefore one obtains a canonical isomorphism (9.12) H p;q (X; C ) ' F p H l DR =F p+1 H l DR ' F p H l DR " F q H l DR ae H l DR : 41 9. the hodge-fr"olicher spectral sequence We deduce from this that there are canonical isomorphisms H l DR (X; C ) = M p+q=l H p;q (X; C ); H q;p (X; C ) = H p;q (X; C ); as expected. Note that (9.12) furnishes another proof of the fact that the Hodge decomposition of a compact K"ahler manifold does not depend on the choice of K"ahler metric (all the groups and morphisms concerned in (9.12) are intrinsic). In fact, we have shown that a compact K"ahler manifold satisfies a still stronger property, that will be convenient to call a strong Hodge decomposition, since this one trivially implies the existence of a Hodge decomposition in the sense of Definition 9.11. 9.13. Definition. If X is a compact complex manifold, we say that X admits a strong Hodge decomposition if the morphisms H p;q BC (X; C ) ! H p;q (X; C ); M p+q=l H p;q BC (X; C ) ! H l DR (X; C ) are isomorphisms. 9.14. Remark. Deligne [Del68, 72] has given an algebraic criterion for the degeneration of the Hodge spectral sequence, including the case of the relative situation. More recently, Deligne and Illusie [DeI87] have given a proof of the degeneration of the Hodge spectral sequence which does not use analytic methods (their idea is to work in characteristic p and to relate the result in characteristic 0). It is necessary to observe that the degeneration of the Hodge-Fr"olicher spectral sequence does not automatically imply the Hodge symmetry property H q;p (X; C ) = H p;q (X; C ) nor the existence of a canonical decomposition of de Rham groups. In fact, it is not difficult to show that the Hodge-Fr"olicher spectral sequence of a compact complex surface always degenerates at E1; however if X is not K"ahler, then b1 is odd, and one can show using the index theorem of Hirzebruch that h 0;1 = h 1;0 + 1 and b1 = 2h 1;0 + 1 (see [BPV84]). One can show that the existence of a Hodge decomposition (resp. strong Hodge) is preserved by contraction morphisms (replacement of X by X 0 , if _ : X ! X 0 is a modification); this is an easy consequence of the existence of a direct image functor _\Lambda acting on all the cohomology groups concerned, such that _\Lambda _ \Lambda = Id. In the analytic context, _\Lambda is easily constructed by calculating cohomology with the aid of currents, since one has on those a natural direct image functor. As any Moishezon manifold admits a projective algebraic modification, we deduce that Moishezon manifolds also admit a strong Hodge decomposition. It would be interesting to know if there exists examples of compact complex manifolds possessing a Hodge decomposition without having a strong Hodge decomposition (there are indeed immediate examples of abstract double complexes having this property). \Lambda In general, when X is not K"ahler, a certain amount of interesting information can be deduced from the spectral sequence. For example, (9.6) implies (9.15) b1 * dim E 1;0 2 + (dim E 0;1 2 \Gamma dim E 2;0 2 )+: In addition, E 1;0 2 is the cohomology group defined by the sequence d1 = d 0 : E 0;0 1 ! E 1;0 1 ! E 2;0 1 ; j.-p. demailly, part i: L 2 hodge theory 42 and since E 0;0 1 is the space of global holomorphic functions on X, the first arrow d1 is zero (by the maximum principal, the holomorphic functions are constant on each connected component of X). Therefore dim E 1;0 2 * h 1;0 \Gamma h 2;0 . Similarly, E 0;1 2 is the kernel of the map E 0;1 1 ! E 1;1 1 , therefore dim E 0;1 2 * h 0;1 \Gamma h 1;1 . From (9.15) we deduce (9.16) b1 * (h 1;0 \Gamma h 2;0 )+ + (h 0;1 \Gamma h 1;1 \Gamma h 2;0 )+: Another interesting relation concerns the topological Euler-Poincar'e characteristic O/top(X) = b0 \Gamma b1 + \Delta \Delta \Delta \Gamma b2n\Gamma 1 + b2n: We utilize the following simple lemma. 9.17. Lemma. Let (C ffl ; d) be a bounded complex of finite dimensional vector spaces over a field. Then the Euler characteristic O/(C ffl ) = X (\Gamma 1) q dim C q is equal to the Euler characteristic O/ \Gamma H ffl (C ffl ) \Delta of the cohomology module. Proof. Set cq = dim C q ; zq = dim Z q (C ffl ); bq = dim B q (C ffl ); hq = dim H q (C ffl ): Then cq = zq + bq+1; hq = zq \Gamma bq: Consequently we find X (\Gamma 1) q cq = X (\Gamma 1) q zq \Gamma X (\Gamma 1) q bq = X (\Gamma 1) q hq: \Lambda In particular, if the term E ffl r of the spectral sequence of a filtered complex K ffl is a bounded complex of finite dimension, one has O/(E ffl r ) = O/(E ffl r+1 ) = \Delta \Delta \Delta = O/(E ffl 1 ) = O/(H ffl (K ffl )) because E ffl r+1 = H ffl (E ffl r ) and dim E l 1 = dim H l (K ffl ). In the Hodge-Fr"olicher spectral sequence one additionally has dim E l 1 = P p+q=l h p;q , therefore: 9.18. Theorem. For any compact complex manifold X, the topological Euler characteristic can be written O/top(X) = X 0^l^2n (\Gamma 1) l bl = X 0^p;q^n (\Gamma 1) p+q h p;q : We now translate the Hodge-Fr"olicher spectral sequence in terms of the spectral sequence of hypercohomology associated to the holomorphic de Rham complex. First let us briefly explain what this spectral sequence consists of. Assume given a bounded complex of sheaves of abelian groups A ffl over a topological space X. Then the hypercohomology groups of A ffl are defined as the groups H k (X; A ffl ) := H k (\Gamma (X; L ffl )); where L ffl is a complex of acyclic sheaves (flasque sheaves or sheaves of C 1 modules for example) chosen so that one has a quasi-isomorphism A ffl ! L ffl (a morphism 43 10. deformations and the semi-continuity theorem of complexes of sheaves inducing an isomorphism H k (A ffl ) ! H k (L ffl ) on the cohomology of sheaves). It is easy to see that hypercohomology does not depend up to isomorphism on the complex of acyclic sheaves L ffl chosen. Hypercohomology is a functor from the category of complexes of sheaves of abelian groups to the category of graded groups. By definition, if A ffl ! B ffl is a quasi-isomorphism, then H k (X; A ffl ) ! H k (X; B ffl ) is an isomorphism; moreover hypercohomology reduces to the usual cohomology H k (X; E) of the sheaf E for a complex A ffl reduced to a single term A 0 = E. Suppose that one has for each term A p of the complex A ffl a resolution A p ! L p;ffl by acyclic sheaves L p;q , giving rise to a double complex of sheaves (L p;q ; d 0 + d 00 ). Then the associated total complex (L ffl ; d) is an acyclic complex quasi-isomorphic to A ffl , and one therefore has H k (X; A ffl ) = H k (\Gamma (X; L ffl )): Further, the double complex K p;q = \Gamma (X; L p;q ) defines a spectral sequence such that E p;q 1 = H q (K p;ffl ; d 00 ) = H q (X; A p ); converges to the associated graded cohomology of the total complex H k (K ffl ) = H k (X; A ffl ). One therefore obtains a spectral sequence called the hypercohomology spectral sequence (9.19) E p;q 1 = H q (X; A p ) ) G p H p+q (X; A ffl ): The filtration F p of hypercohomology groups is by definition obtained by taking the image of the morphism H k (X; F p A ffl ) ! H k (X; A ffl ); where F p A ffl denotes the complex truncated to the left \Delta \Delta \Delta ! 0 ! 0 ! A p ! A p+1 ! \Delta \Delta \Delta ! A N \Delta \Delta \Delta : Consider now the case where X is any given complex manifold and where A ffl = \Omega ffl X is the holomorphic de Rham complex (with the usual exterior differential). The holomorphic Poincar'e Lemma shows that \Omega ffl X is a resolution of the constant sheaf C X , i.e., one has a quasi-isomorphism of complexes of sheaves C X ! \Omega ffl X , where C X denotes the complex reduced to a single term in degree 0. By definition of hypercohomology, one therefore has (9.20) H k (X; C X ) = H k (X; \Omega ffl X ); and the exact sequence of hypercohomology of the complex \Omega ffl X furnishes a spectral sequence (9.21) E p;q 1 = H q (X; \Omega p X ) ) G p H p+q (X; C X ): Because the groups H k (X; \Omega ffl X ) can be calculated by using the resolution of \Omega ffl X by the Dolbeault complex L p;q = C 1 (\Lambda p;q T \Lambda X ) (these sheaves are certainly acyclic!), one then sees that the hypercohomology spectral sequence (9.21) is precisely the Hodge-Fr"olicher spectral sequence previously defined. j.-p. demailly, part i: L 2 hodge theory 44 10. Deformations and the semi-continuity theorem The purpose of this section is to study the dependence of the groups H p;q (Xt; C ) or more generally the cohomology groups H q (Xt; Et), when the pair (Xt; Et) depends holomorphically on a parameter t in a certain complex space S. Our approach is to adopt the point of view of Kodaira-Spencer, such as is developed in their original work on the theory of deformations (see for example the complete works of Kodaira [Kod75]). The method of Kodaira-Spencer exploits the continuity properties or semi-continuity of proper spaces of Laplacians as a function of the parameter t. Another approach furnishing more precise results consists of utilizing the theorem of direct images of Grauert [Gra60]. 10.1 Definition. A deformation of compact complex manifolds is given by a proper analytic morphism oe : X ! S of connected complex spaces, for which all the fibers Xt = oe \Gamma 1 (t) are smooth manifolds of the same dimension n, and satisfy the following local condition: (H) Any point i 2 X admits a neighbourhood U such that there exists a biholomorphism : U \Theta V ! U where U is open in C n and V is a neighbourhood of t = oe(i), satisfying oe ffi = pr2 : U \Theta V ! V (second projection). We say that (Xt)t2S is a holomorphic family of deformations of any given fiber Xt 0 , and that S is the base of the deformation. A holomorphic family of vector bundles (resp. sheaves) Et ! Xt is given by a family of bundles (resp. sheaves) obtained from a global bundle (resp. global sheaf) E ! X, by restriction to the fibers Xt. If S is smooth, the hypothesis (H) is equivalent to assuming that oe is a holomorphic submersion, as a consequence of the theorem of constant rank. There are nevertheless situations where one must necessarily consider also the case of a singular base S (for example when one seeks to construct the "universal deformation" of a manifold). In a topological setting (differentiable or smooth), we have the following lemma, known as Ehresmann's Lemma. 10.2. Ehresmann's Lemma. Let oe : X ! S be a smooth and proper differentiable submersion. a) If S is contractible, then for any t0 2 S, there exists a commutative diagram X \Phi \Gamma ! Xt 0 \Theta S pr1 & . oe S where \Phi is a diffeomorphism. b) For any given base S, X ! S is a locally trivial bundle (differentiable). In particular, if S is connected, the fibers are all diffeomorphic. Proof. a) Let H : S \Theta [0; 1] ! S be a differentiable homotopy between H(ffl; 0) = IdS and H(ffl; 1) = constant map S ! ft0g. The fiber product ~ X = f(x; s; t) 2 X \Theta S \Theta [0; 1] ; oe(x) = H(s; t)g with projection ~oe = pr2 \Theta pr3 : ~ X ! S \Theta [0; 1] is still a differentiable submersion, as one can easily verify. One deduces that there exists a vector field , on ~ X which lifts the vector field @ @t on S \Theta [0; 1], i.e. oe\Lambda , = @ @t . (There exists a local lifting by the 45 10. deformations and the semi-continuity theorem submersive property, and one glues together these liftings by means of a partition of unity.) Let 't be a flow of this lifting: Then, if (x; s; 0) 2 ~ X_S\Theta f0g ' X, one has by construction 't(x; s; 0) = (?; s; t), therefore \Phi = '1 defines a diffeomorphism of ~ X_S\Theta f0g ' X on ~ X_S\Theta f1g ' Xt 0 \Theta S, commuting with the projection on S. b) is deduced immediately from a). \Lambda It follows from b) that the bundle t 7! H k (Xt; C ) is a locally trivial bundle of C vector spaces of finite dimension. Furthermore, in each fiber we have a free abelian subgroup ImH k (Xt; Z) ae H k (Xt; C ) of rank bk which generates H k (Xt; C ) as a C vector space. The transition matrices of this locally constant system are in SLb k (Z). Since the transition matrices are locally constant, the bundle t 7! H k (Xt; C ) is equipped with a connection D such that D 2 = 0: This connection is called the Gauss-Manin connection. The following lemma is useful. 10.3. Lemma. Let oe : X ! S be a smooth and proper differentiable submersion and E a C 1 vector bundle over X. Consider a family of elliptic operators Pt : C 1 (Xt; Et) ! C 1 (Xt; Et) of degree ffi. We assume that Pt is self-adjoint semipositive relative to a metric ht on Et and a volume form dVt on Xt, and that the coefficients of Pt, ht and dVt are C 1 on X. Then the eigenvalues of Pt, computed with multiplicity, can be arranged in a sequence *0(t) ^ *1(t) ^ \Delta \Delta \Delta ^ *k(t) ! +1; where the k-th eigenvalue *k(t) is a continuous function of t. Moreover, if * is not in the spectrum f*k(t0)gk2N of Pt 0 , the direct sum W*;t ae C 1 (Xt; Et) of eigenspaces of Pt with eigenvalues *k(t) ^ * defines a C 1 vector bundle, t 7! Wt;*, in a neighbourhood of t0. Proof. Since the results are local over S, one can assume that X = Xt 0 \Theta S and E = pr \Lambda 1 Et 0 , that is, their fibers Xt and Et are independent of t (but the forms dVt on Xt and the metrics ht on Et are in general dependent on t). Let \Pi *;t be the orthogonal projection operator on W*;t in L 2 (Xt; Et) ' L 2 (Xt 0 ; Et 0 ). If \Gamma (0; *) denotes the circle with center 0 and with radius * in the complex plane, Cauchy's formula gives \Pi *;t = 1 2ssi Z \Gamma (0;*) (zId \Gamma Pt) \Gamma 1 dz; where the integral is viewed as an integral with vector values in the space of bounded operators on L 2 (Mt 0 ; Et 0 ). (It suffices to verify the formula on the eigenvectors of Pt, which is elementary.) The arguments made in x3 show that there exists a family of pseudodifferential operators Qt of order \Gamma ffi, for which the symbol depends in a C 1 manner with t (and with uniform estimates by differentiation in t), such that PtQt = Id + Rt for regular operators Rt, for which the kernel also depends in a C 1 manner in t. Since Qt is a family of compact operators on L 2 (Xt; Et) which depend in a C 1 manner in t, the eigenvalues of Qt depend continuously in t. Up to changing Qt 0 on a subspace of finite dimension, one can assume that Qt 0 is an isomorphism of L 2 (Xt 0 ; Et 0 ) onto W ffi (Xt 0 ; Et 0 ). It will be the same for Qt in a neighbourhood of t0, and consequently zId\Gamma Pt is invertible if and only if (zId\Gamma Pt)Qt = Id+Rt +zQt is invertible. If * is not in the spectrum of Pt 0 , it follows that for all z 2 \Gamma (0; *), the inverse (zId \Gamma Pt) \Gamma 1 = Qt(Id + Rt + zQt) \Gamma 1 depends in a C 1 way in t. This j.-p. demailly, part i: L 2 hodge theory 46 implies that t 7! Wt;* is a locally trivial C 1 fibration in a neighbourhood of t0. The continuity of the eigenvalue *k(t) of Pt follows from the constant rank of W*;t in a neighbourhood of t0, for * = *k(t0) \Sigma ffl. \Lambda 10.4. Semi-continuity Theorem (Kodaira-Spencer). If X ! S is a smooth, proper C -analytic morphism and if E is a locally free sheaf on X, the dimensions h q (t) = h q (Xt; Et) are upper semi-continuous functions. More precisely, the alternating sums h q (t) \Gamma h q\Gamma 1 (t) + \Delta \Delta \Delta + (\Gamma 1) q h 0 (t); 0 ^ q ^ n = dim Xt are upper semi-continuous functions. Proof. Let Et the holomorphic vector bundle associated to Et. Equip E and X with arbitrary Hermitian metrics. According to the Hodge isomorphism for the d 00 -cohomology, one can interpret H q (Xt; Et) as the space of harmonic forms for the Laplacian \Delta 00q t acting on C 1 (Xt; \Lambda 0;q T \Lambda Xt \Omega Et). Fix a point t0 2 S and a real * ? 0 which does not belong in the spectrum of the operators \Delta 00q t0 ; 0 ^ q ^ n = dim Xt. Then W q t = W q *;t = direct sum of eigenspaces of \Delta 00q t with eigenvalues ^ * defines a C 1 bundle W q in a neighbourhood of t0. Moreover the differential d 00 t commutes with \Delta 00 t and thus sends the eigenspaces of \Delta 00q t into the eigenspaces of \Delta 00q+1 t associated to the same eigenvalues. This shows that (W ffl t ; d 00 t ) is a subcomplex of finite dimension of the Dolbeault complex \Gamma C 1 (Xt; \Lambda 0;q T \Lambda Xt \Omega Et); d 00 t \Delta . The cohomology of this subcomplex coincides with H q (Xt; Et) since the relation d 00 t d 00\Lambda t + d 00\Lambda t d 00 t = \Delta 00 t shows that 1 *k d 00\Lambda t is a homotopy operator on the subcomplex formed from the eigenspaces with eigenvalue *k when *k 6= 0. If Z q t denotes the kernel of the morphism d 00q t : W q t ! W q+1 t , then z q (t) := dim Z q t is an upper semicontinuous function in the Zariski topology, as one can easily see by considering the rank of the minors of the matrix defining the morphism d 00q : W q ! W q+1 . From the truncated complex 0 ! W 0 t ! W 1 t ! \Delta \Delta \Delta ! W q\Gamma 1 t ! Z q t ! 0 having for the cohomology the groups H j (Xt; Et) with indices 0 ^ j ^ q, one obtains h q (t) \Gamma h q\Gamma 1 (t) + \Delta \Delta \Delta + (\Gamma 1) q h 0 (t) = z q (t) \Gamma w q\Gamma 1 + w q\Gamma 2 + \Delta \Delta \Delta + (\Gamma 1) q w 0 ; where w q denotes the rank of W q . The upper semi-continuity of the term on the left follows, and that of h q (t) is then immediate by induction on q. \Lambda 10.5. Invariance of the Hodge numbers. Let X ! S be a smooth and proper C -analytic morphism. We assume that the fibers Xt are K"ahler manifolds. Then the Hodge numbers h p;q (Xt) are constant. Moreover, in the decomposition H k (Xt; C ) = M p+q=k H p;q (Xt; C ); the bundles t 7! H p;q (Xt; C ) define C 1 subbundles (in general, not holomorphic subbundles) of the bundle t 7! H k (Xt; C ). 47 10. deformations and the semi-continuity theorem Proof. Lemma 10.2 implies that the Betti numbers bk = dim H k (Xt; C ) are constant. Since, according to th. 10.4, h p;q (Xt; C ) = h q (Xt; \Omega p Xt ) is upper semicontinuous, and h p;q (Xt) = bk \Gamma X r+s=k;(r;s)6=(p;q) h r;s (Xt); these functions are likewise lower semi-continuous. Consequently they are continuous and therefore constant. A theorem of Kodaira [Kod75] shows that if a fiber Xt 0 is K"ahler, then the neighbouring fibers Xt are K"ahler and the K"ahler metrics !t can be chosen so that they depend in a C 1 way with t. The spaces of harmonic (p; q)-forms therefore depend in a C 1 way with t according to th. 10.4, and one deduces that t 7! H p;q (Xt; C ) is a C 1 subbundle of H k (Xt; C ). \Lambda It is possible to obtain more precise and general results by means of the theorem of direct images of Grauert [Gra60]. Recall that if we are given a continuous map f : X ! Y between topological spaces and a sheaf E of abelian groups on X, then one can define the direct image sheaf R k f\Lambda E on Y , as being the sheaf associated to the presheaf U 7! H k (f \Gamma 1 (U ); E), for all open U in Y . More generally, being given a complex of sheaves A ffl , we have the direct image sheaves R q f\Lambda A ffl , obtained from the hypercohomology presheaves U 7! H k (f \Gamma 1 (U ); A ffl ): The proof of the theorem of direct images as given by [FoK71] and [KiV71] (also see [DoV72]) furnishes the following fundamental result. 10.6. Theorem of direct images. Let oe : X ! S be a proper morphism of complex analytic spaces and A ffl a bounded complex of coherent sheaves of OXmodules. Then a) The direct image sheaves R k oe\Lambda A \Lambda are coherent sheaves on S. b) Any point of S admits a neighbourhood U ae S on which there exists a bounded complex W ffl of sheaves of locally free OS-modules in which the cohomology sheaves H k (W ffl ) are isomorphic to the sheaf R k oe\Lambda A ffl . c) If the fibers of oe are equidimensional ("geometrically flat morphism"), the hypercohomology of the fiber Xt = oe \Gamma 1 (t) with values in A ffl t = A ffl \Omega O X OX t (where OX t = OX=oe \Lambda mS;t) is given by H k (Xt; A ffl t ) = H k (W ffl t ); where (W ffl t ) is the complex of finite dimensional spaces W k t = W k \Omega O S;t OS;t=mS;t. d) Under the hypothesis of c), if the hypercohomology spaces H k (Xt; A ffl t ) of the fibers are of constant dimension, the sheaves R k oe\Lambda A ffl are locally free on S. The same results are true in particular for the direct images R k oe\Lambda E of a coherent sheaf E on X, and the cohomology groups H k (Xt; Et) of the fibers. One notes that property d) is in fact a formal consequence of c), because the hypothesis guarantees that the holomorphic matrices defining morphisms W k ! W k+1 are of constant rank at each point t 2 S. From (10.6b) one then deduces the following result due to [Fle81] with an identical argument to that in th. 10.4. 10.7 Semi-Continuity Theorem. If X ! S is a proper analytic morphism with equidimensional fibers and if E is a coherent sheaf on X, then the alternating j.-p. demailly, part i: L 2 hodge theory 48 sums h q (t) \Gamma h q\Gamma 1 (t) + \Delta \Delta \Delta + (\Gamma 1) q h 0 (t); with dimensions h k (t) = h k (Xt; Et), are upper semi-continuous functions of t in the analytic Zariski topology (topology of whose closed are the analytic sets). Let oe : X ! S be a C -analytic proper and smooth submersion. One assumes that the Hodge spectral sequence of the fibers Xt degenerates at E1 for all t 2 S (according to (10.7) this is in fact an open property for the analytic Zariski topology on S). If U ae S is open and contractible, then oe \Gamma 1 (U ) ' Xt \Theta U for any fiber over t 2 U . If ZX; C X , denotes the locally constant sheaves with base X and with fibers Z; C , one obtains \Gamma (U; R k oe\Lambda ZX) = H k (oe \Gamma 1 (U ); Z) = H k (Xt; Z); \Gamma (U; R k oe\Lambda C X ) = H k (oe \Gamma 1 (U ); C ) = H k (Xt; C ); so that R k oe\Lambda ZX and R k oe\Lambda C X are locally constant sheaves on S, with fibers H k (Xt; Z) and H k (Xt; C ). The bundle t 7! H k (Xt; C ), equipped with the flat connection D (Gauss-Manin connection), possesses a canonical holomorphic structure induced by the component D 0;1 of the Gauss-Manin connection. The flat bundle \Phi kH k (Xt; C ) is called the Hodge bundle of the fibration X ! S. Now consider the relative de Rham complex (\Omega ffl X=S ; dX=S) of the fibration X ! S. This complex furnishes a resolution of the sheaf oe \Gamma 1 OS ("purely sheafified" inverse image of OS), consequently (10.8) R k oe\Lambda \Omega ffl X=S = R k oe\Lambda (oe \Gamma 1 OS) = (R k oe\Lambda C X ) \Omega C OS: The latter equality is obtained immediately by an argument using OS(U ) linearity for the cohomology calculated on the open set oe \Gamma 1 (U ) (the complex structure of oe \Gamma 1 (U ) does not intervene here). In other words, R k oe\Lambda \Omega ffl X=S is the locally free OS-module associated to the flat bundle t 7! H q (Xt; C ). One has a relative hypercohomology spectral sequence E p;q 1 = R q oe\Lambda \Omega p X=S ) G p R p+q oe\Lambda \Omega ffl X=S = G p R p+q oe\Lambda C X (the relative spectral sequence is obtained simply by a "sheafification" of the absolute hypercohomology spectral sequence (9.19) of the complex \Omega ffl X=S over the open set oe \Gamma 1 (U )). Since the cohomology of \Omega p X=S on the fiber Xt is precisely the space H q (Xt; \Omega p Xt ) of constant rank, th. 10.6d) shows that the direct image sheaves R p oe\Lambda \Omega p X=S are locally free. In addition, the filtration F p H k (Xt; C ) ae H k (Xt; C ) is obtained on the level of locally free OS-modules associated with taking the image of the OS-linear morphism R k oe\Lambda F p \Omega ffl X=S ! R k oe\Lambda \Omega ffl X=S ; which is therefore a coherent subsheaf (and likewise a locally free subsheaf, according to the property of constant rank on the fibers Xt). From (10.8) one deduces the 10.9. Theorem (holomorphic Hodge filtration). The Hodge filtration F p H k (Xt; C ) ae H k (Xt; C ) defines a holomorphic subbundle relative to the holomorphic structure defined by the Gauss-Manin connection. One sees that in general there is no reason for H p;q (Xt; C ) = F p H k (Xt; C ) " F q H k (Xt; C ) to be a holomorphic subbundle of H k (Xt; C ) for any p + q = k, although H p;q (Xt; C ) possesses a natural holomorphic bundle structure (obtained from the coherent sheaf R q oe\Lambda \Omega p X=S , or as a quotient of F p H k (Xt; C )). In other words, this is the Hodge decomposition which is not holomorphic. 10.10 Example. Let S = fo/ 2 C ; Im o/ ? 0g be the upper half plane and X ! S the "universal" family of elliptic curves over S, defined by Xo/ = C =(Z + Zo/ ). The two basis elements of the Hodge fiber H 1 (Xo/ ; C ), dual to the basis (1; o/ ) of the lattice of periods, are ff = dx \Gamma Re o/ =Im o/ dy and fi = (Im o/ ) \Gamma 1 dy (z = x + iy 2 C denotes the coordinates on Xo/ ). These elements therefore satisfy Dff = Dfi = 0 and define the holomorphic structure of the Hodge bundle; the subbundle H 1;0 (Xo/ ; C ) generated by the 1-form dz = ff + o/ fi is clearly holomorphic (as it should be!), however one sees that the components fi 1;0 = \Gamma i 2 (Im o/ ) \Gamma 1 dz and fi 0;1 = \Gamma i 2 (Im o/ ) \Gamma 1 dz are not holomorphic in o/ . 49 j.-p. demailly, part ii: L 2 estimations and vanishing theorems 50 Part II: L 2 Estimations and Vanishing Theorems 11. Concepts of pseudoconvexity and of positivity The statements and proofs of the vanishing theorems brings into play many concepts of pseudoconvexity and positivity. We first present a summary, by bringing together the concepts that we deem necessary. 11.A. Plurisubharmonic functions. The plurisubharmonic functions were introduced independently by Lelong and Oka in 1942 in the study of holomorphic convexity. We refer to [Lel67, 69] for more details. 11.1. Definition. A function u : \Omega ! [\Gamma 1; +1[ defined on an open set \Omega ae C n is called plurisubharmonic (abbreviated psh) if a) u is upper semi-continuous; b) for any complex line L ae C n ; u_\Omega "L is subharmonic on \Omega " L, that is, for any a 2 \Omega and , 2 C n satisfying j,j ! d(a; -\Omega ), the function u satisfies the mean inequality u(a) ^ 1 2ss Z 2ss 0 u(a + e i\Theta ,)d`: The set of psh functions on \Omega is denoted by Psh(\Omega ). We give below a list of some fundamental properties satisfied by the psh functions. All these properties come about easily from the definition. 11.2. Fundamental properties. a) Any function u 2 Psh(\Omega ) is subharmonic in the 2n real variables, i.e. satisfies the mean value inequality on the Euclidean ball (or sphere): u(a) ^ 1 ss n r 2n =n! Z B(a;r) u(z)d*(z) for all a 2 \Omega and all r ! d(a; -\Omega ). In this case, one has either u j \Gamma 1 or u 2 L 1 loc on every connected component of \Omega . b) For any decreasing sequence of psh functions uk 2 Psh(\Omega ), the limit u = lim uk is psh on \Omega . c) Assume given u 2 Psh(\Omega ) such that u 6j \Gamma 1 on all connected components of \Omega . If (aeffl) is a family of regular kernels, then u ? aeffl is C 1 and psh on \Omega ffl = fx 2 \Omega ; d(x; -\Omega ) ? fflg; the family (u ? aeffl) is increasing in ffl, and limffl!0 u ? aeffl = u. d) Assume given u1; : : : ; up 2 Psh(\Omega ) and O/ : R p ! R a convex function such that O/(t1; : : : ; tp) is increasing in each variable tj. Then O/(u1; : : : ; up) is psh on \Omega . In particular u1 + \Delta \Delta \Delta + up; maxfu1; : : : ; upg; log(e u1 + \Delta \Delta \Delta + e up ) are psh on \Omega . \Lambda 11.3. Lemma. A function u 2 C 2 (\Omega ; R) is psh on \Omega if and only if the Hermitian form Hu(a)(,) = P 1^j;k^n @ 2 u=@zj@zk(a),j , k is semi-positive at every point a 2 \Omega . 51 11. concepts of pseudoconvexity and of positivity Proof. This is an easy consequence of the following standard formula 1 2ss Z 2ss 0 u(a + e i\Theta ,)d` \Gamma u(a) = 2 ss Z 1 0 dt t Z jij!t Hu(a + i,)(,)d*(i); where d* is the Lebesque measure on C . Lemma 11.3 strongly suggests that plurisubharmonicity is the complex analog of the property of linear convexity in the real case. \Lambda For nonregular functions, one obtains an analogous characterization of plurisubharmonicity by means of a process of regularization. 11.4. Theorem. If u 2 Psh(\Omega ) with u 6j \Gamma 1 on every connected component of \Omega , then for every , 2 C n Hu(,) = X 1^j;k^n @ 2 u @zj@zk ,j , k 2 D 0 (\Omega ) is a positive measure. Conversely, if v 2 D 0 (\Omega ) is given such that Hv(,) is a positive measure for all , 2 C n , then there exists a unique function u 2 Psh(\Omega ) which is locally integrable on \Omega and such that v is the distribution associated to u. \Lambda In order to obtain a better geometrical comprehension of the notion of plurisubharmonicity, we assume more generally that the function u lives on a complex manifold X of dimension n. If \Phi : X ! Y is a holomorphic map and if v 2 C 2 (Y; R), we have d 0 d 00 (v ffi \Phi ) = \Phi \Lambda d 0 d 00 v, therefore H(v ffi \Phi )(a; ,) = Hv(\Phi (a); \Phi 0 (a):,): In particular Hu, viewed as a Hermitian form on TX, is independent of the choice coordinates (z1; : : : ; zn). Consequently, the notion of a psh function makes sense on any complex manifold. More generally, we have 11.5. Proposition. If \Phi : X ! Y is a holomorphic map and v 2 Psh(Y ), then v ffi \Phi 2 Psh(X). \Lambda 11.6. Example. It is well known that log jzj is psh (i.e. subharmonic) on C . Therefore log jf j 2 Psh(X) for any holomorphic function f 2 H 0 (X; OX). More generally log(jf1j ff1 + \Delta \Delta \Delta + jfqj ffq ) 2 Psh(X) for any choice of functions fj 2 H 0 (X; OX ) and real ffj * 0 (apply property 11.2d with uj = ffj log jfjj). We will be interested more particularly with singularities of this function along the variety of zeros f1 = \Delta \Delta \Delta = fq = 0, when the ffj are rational numbers. \Lambda 11.7. Definition. One says that a psh function u 2 Psh(X) has analytic singularities (resp. algebraic) if u can be written locally in the form u = ff 2 log(jf1j 2 + \Delta \Delta \Delta + jfN j 2 ) + v; with holomorphic functions (resp. algebraic) fj; ff 2 R+ , (resp. ff 2 Q+ ), and where v is a bounded function. j.-p. demailly, part ii: L 2 estimations and vanishing theorems 52 We introduce then the ideal J = J(u=ff) of germs of holomorphic functions h such that there exists a constant C * 0 for which jhj ^ Ce u=ff , i.e. jhj ^ C(jf1j + \Delta \Delta \Delta + jfN j): One therefore obtains a global sheaf of ideals defined on X, locally equal to the integral closure I of the sheaf of ideals I = (f1; : : : ; fN ); consequently J is coherent on X. If (g1; : : : ; gN0) are the local generators of J, we still have u = ff 2 log(jg1j 2 + \Delta \Delta \Delta + jgN0j 2 ) + O(1): From an algebraic point of view, the singularities of u are in bijective correspondence with the "algebraic data" (J; ff). We will later see another even more significant way to associate to a psh function, a sheaf of ideals. 11.B. Positive currents. The theory of currents was founded by G. de Rham [DR55]. We mention here only the most basic definitions. The reader can consult [Fed69] for a much more complete treatment of this theory. In the complex situation, the important characteristic concept of a positive current was studied and emanated by P. Lelong [Lel57,69]. A current of degree q on a differential manifold M , is nothing more than a differential q-form \Theta with distribution coefficients. The space of currents of degree q on M will be denoted by D 0q (M ). Alternatively, one can consider the currents of degree q as the elements \Theta of the dual D 0 p (M ) := \Gamma D p (M ) \Delta 0 of the space D p (M ) of C 1 differential forms of degree p = dim M \Gamma q with compact support; the duality pairing is given by (11.8) h\Theta ; ffi = Z M \Theta ^ ff; ff 2 D p (M ): A fundamental example is the current of integration [S] on a compact oriented submanifold S (possibly with boundary) of M : (11.9) h[S]; ffi = Z S ff; deg ff = p = dimR S: Then [S] is a current with measurable coefficients, and Stokes theorem shows that d[S] = (\Gamma 1) q\Gamma 1 [@S]. In particular d[S] = 0 if and only if S is a submanifold without boundary. Because of this example, the integer p is called the dimension of \Theta 2 D 0 p (M ). One says that the current \Theta is closed if d\Theta = 0. On a complex manifold X, we have the analogous concept of bidegree and of bidimension. As in the real case, we denote by D 0p;q (X) = D 0 n\Gamma p;n\Gamma q (X); n = dim X; the space of currents of bidegree (p; q) and bidimension (n\Gamma p; n\Gamma q) on X. Following [Lel57], a current \Theta of bidimension (p; p) is called (weakly) positive if for any choice of C 1 (1; 0)-forms ff1; : : : ; ffp on X, the distribution (11.10) \Theta ^ iff1 ^ ff1 ^ \Delta \Delta \Delta ^ iffp ^ ffp is a positive measure. 53 11. concepts of pseudoconvexity and of positivity 11.11. Exercise. If \Theta is positive, show that the coefficients \Theta I;J of \Theta are complex measures, and that they are dominated up to a constant by the trace measure oe\Theta = \Theta ^ 1 p! fi p = 2 \Gamma p X \Theta I;I ; where fi = i 2 d 0 d 00 jzj 2 = i 2 X 1^j^n dzj ^ dzj; is a positive measure. Indication. Observe that P \Theta I;I is invariant under a unitary change of coordinates, and that the (p; p)-forms iff1 ^ ff1 \Delta \Delta \Delta ^ iff ^ ffp generate \Lambda p;p T \Lambda Cn as a C -vector space. \Lambda One easily sees that a current \Theta = i P 1^j;k^n \Theta jkdzj ^ dzk of bidegree (1; 1) is positive if and only if the complex measure P *j*k\Theta jk is a positive measure for any n-tuple (*1; : : : ; *n) 2 C n . 11.12. Example. If u is a psh function (not identically \Gamma 1) on X, one can associate to u a closed positive current \Theta = i@@u of bidegree (1; 1). Conversely, any closed positive current of bidegree (1; 1) can be written in this form on any open subset \Omega ae X satisfying H 2 DR (\Omega ; R) = H 1 (\Omega ; O) = 0, for example on open coordinate charts biholomorphic to a ball (exercise for the reader). \Lambda It is not difficult to show that a product \Theta 1 ^ \Delta \Delta \Delta ^ \Theta q of positive currents of bidegree (1; 1) is positive whenever the product is well defined. (This is certainly the case if all but one of the \Theta j are C 1 .) Other much finer conditions exist, but we will not pursue this subject here. We now discuss another very important example of a closed positive current. For any closed analytic set A in X, of pure dimension p, one associates a current of integration (11.13) h[A]; ffi = Z Areg ff; ff 2 D p;p (X); obtained by integrating ff on the set of regular points of A. To check that (11.13) gives a legitimate definition of a current on X, it should be shown that Areg is locally of finite area in a neighbourhood of each point of Asing. This result which due to [Lel57], can be shown as follows. Suppose (after a change of coordinates) that 0 2 Asing. From the local parameterization theorem for analytic sets, one deduces that there exists a linear change of coordinates on C n such that all the projections ssI : (z1; : : : ; zn) 7! (zi 1 ; : : : ; zi p ) define a finite ramified covering over the intersection A " \Delta I of A with a small polydisk \Delta I = \Delta 0 I \Theta \Delta 00 I of C n = C p \Theta C n\Gamma p , over the polydisk \Delta 0 I of C p . Let nI be the number layers of each of these coverings. Then, if \Delta = "\Delta I , the p-dimensional area of A " \Delta is bounded above by the sum of the areas of its projections computed with multiplicities, i.e. Surface Area(A " \Delta ) ^ X nIVol(\Delta 0 I ): The fact that [A] is positive is easy. In fact, in terms of local coordinates (w1; : : : ; wp) on Areg, one has iff1 ^ ff1 ^ \Delta \Delta \Delta ^ iffp ^ ffp = j det(ffjk)j 2 iw1 ^ w1 ^ \Delta \Delta \Delta iwp ^ wp j.-p. demailly, part ii: L 2 estimations and vanishing theorems 54 if ffj = P ffjkdwk. This shows that such a product of forms is * 0 by comparison to the canonical orientation defined by iw1 ^ w1 ^ \Delta \Delta \Delta ^ iwp ^ wp. A deeper result, also proven by P. Lelong [Lel57], is that [A] is a d-closed current on X, in other words, the set Asing (which is of real dimension ^ 2p \Gamma 2) does not contribute to the boundary current d[A]. Finally, in connection with example 11.12, we have the important 11.14. Lelong-Poincar'e equation. Let f 2 H 0 (X; OX) be a nonzero holomorphic function, Zf = P mjZj; mj 2 N, the divisor of zeros of f , and [Zf ] = P mj[Zj] the associated current of integration. Then i ss @@ log jf j = [Zf ]: Proof (outline). It is clear that id 0 d 00 log jf j = 0 in a neighbourhood of each point x 62 Supp(Zf ) = [Zj, consequently it suffices to verify the equation in a neighbourhood of any point of Supp(Zf ). Let A be the set of singular points of Supp(Zf ), i.e. the union of the intersections Zj "Zk and of their singularities Zj;sing; we then have dim A ^ n \Gamma 2. In a neighbourhood of any point x 2 Supp(Zf )nA there exists local coordinates (z1; : : : ; zn) such that f (z) = z mj 1 , where mj is the multiplicity of f along the component Zj which contains x, and where z1 = 0 is a local equation of Zj near x. Since i ss d 0 d 00 log jzj = Dirac measure ffi0 in C , we find i ss d 0 d 00 log jz1j = [hyperplane z1 = 0], therefore i ss d 0 d 00 log jf j = mj i ss d 0 d 00 log jz1j = mj[Zj] in a neighbourhood of x. This shows that the equation is valid on XnA. Consequently, the difference i ss d 0 d 00 log jf j \Gamma [Zf ] is a closed current of degree 2 with measurable coefficients for which the support is contained in A. This current is necessarily zero because A is of too small a dimension for to be able to carry its support. (A is stratified into submanifolds of real codimension * 4, whereas the current itself is of real codimension 2.) \Lambda To conclude this section we now revisit the de Rham and Dolbeault cohomology in the context of the theory of currents. A basic observation is that the Poincar'e and Dolbeault-Grothendieck Lemmas are still valid for currents. More precisely, if (D 0q ; d) and (D 0 (F ) p;q ; d 00 ) denotes the complexes of sheaves of currents of degree q (resp. currents of bidegree (p; q) with values in a holomorphic vector bundle F ), one still has resolutions of de Rham and of Dolbeault sheaves 0 ! R ! D 0ffl ; 0 ! \Omega p X \Omega O(F ) ! D 0 (F ) p;ffl : As a result, there are canonical isomorphisms H q DR (M; R) = H q \Gamma (\Gamma (M; D 0ffl ); d) \Delta ;(11.15) H p;q (X; F ) = H q \Gamma (\Gamma (X; D 0 (F ) p;ffl ); d 00 ) \Delta : In other words, one can attach a cohomology class f\Theta g 2 H q DR (M; R) to any closed current \Theta of degree q, resp. a cohomology class f\Theta g 2 H p;q (X; F ) for any d 00 -closed current of bidegree (p; q). By replacing if necessary the respective currents by their 55 11. concepts of pseudoconvexity and of positivity C 1 representatives of the same cohomology class, one sees that there exists a well defined cup product pairing, given by the exterior product of differential forms H q1 (M; R) \Theta \Delta \Delta \Delta \Theta H qm (M; R) ! H q1+\Delta \Delta \Delta +qm (M; R); (f\Theta 1g; : : : ; f\Theta 1g) 7! f\Theta 1g ^ \Delta \Delta \Delta ^ f\Theta mg: In particular, if M is a compact oriented manifold and if q1 + \Delta \Delta \Delta + qm = dim M , one obtains a well defined intersection number f\Theta 1g \Delta f\Theta 2g \Delta \Delta \Delta \Delta \Delta f\Theta mg = Z M f\Theta 1g ^ \Delta \Delta \Delta ^ f\Theta mg: We note however that the specific product \Theta 1 ^ \Delta \Delta \Delta ^ \Theta m does not exist in general. 11.C. Positive vector bundles. Let (E; h) be a Hermitian holomorphic vector bundle on a complex manifold X. Its Chern curvature tensor \Theta (E) = X 1^j;k^n;1^*;_^r cjk*_dzj ^ dzk \Omega e \Lambda * \Omega e_ can be identified with a Hermitian form on TX \Omega E, viz. (11.16) ~ \Theta (E)(, \Omega v) = X 1^j;k^n;1^*;_^r cjk*_,j, k v*v_; cjk*_ = ckj_*: This leads us naturally to the concept of positivity, in the following definitions introduced by Kodaira [Kod53], Nakano [Nak55] and Griffiths [Gri66]. 11.17. Definition. The Hermitian holomorphic vector bundle E is called a) positive in the sense of Nakano if: ~ \Theta (E)(o/ ) ? 0 for any non-zero tensor o/ = P o/j*@=@zj \Omega e* 2 TX \Omega E. b) positive in the sense of Griffiths if: ~ \Theta (E)(, \Omega v) ? 0 for any non-zero decomposable tensor , \Omega v 2 TX \Omega E. The corresponding concepts of semi-positivity are defined by replacing the strict inequalities by the broader inequalities. 11.18. The particular case of rank 1 bundles. Suppose that E is a line bundle. The Hermitian matrix H = (h11) associated to a trivialization o/ : E_\Omega ' \Omega \Theta C is then simply a positive function, and it will be convenient to denote it by e \Gamma 2' , ' 2 C 1 (\Omega ; R). In this case, the curvature form \Theta (E) can be identified with the (1; 1)-form 2d 0 d 00 ', and i 2ss \Theta (E) = i ss d 0 d 00 ' = dd c '; where d c = i 2ss (d 00 \Gamma d 0 ) is a real (1; 1)-form. Therefore E is semipositive (in the sense of Nakano or in the sense of Griffiths) if and only if ' is psh, resp. positive if and only if ' is strictly psh. In this context, the Lelong-Poincar'e equation can be generalized as follows: Let oe 2 H 0 (X; E) be a non-zero holomorphic section. Then (11.19) dd c log jjoejj = [Zoe] \Gamma i 2ss \Theta (E): Formula (11.19) is immediate if one writes jjoejj = jo/ (oe)je \Gamma ' and if one applies the Lelong-Poincar'e equation to the holomorphic function f = o/ (oe). As we will see later, it is important for applications to consider the case of singular Hermitian metrics (cf. [Dem90b]). j.-p. demailly, part ii: L 2 estimations and vanishing theorems 56 11.20. Definition. A singular (Hermitian) metric on a line bundle E is a metric given in any trivialization o/ : E_\Omega ' \Gamma ! \Omega \Theta C by jj,jj = jo/ (,)je \Gamma '(x) ; x 2 \Omega ; , 2 Ex where ' 2 L 1 loc (\Omega ) is an arbitrary function, called the weight of the metric with respect to the trivialization o/ . If o/ 0 : E_\Omega 0 ! \Omega 0 \Theta C is another trivialization, ' 0 the associated weight and g 2 O \Lambda (\Omega " \Omega 0 ) the transition function, then o/ 0 (,) = g(x)o/ (,) for all , 2 EX , and therefore ' 0 = ' + log jgj on \Omega " \Omega 0 . The curvature form of E is then formally given by the current of degree (1; 1), i ss \Theta (E) = dd c ' on \Omega ; moreover the hypothesis ' 2 L 1 loc (\Omega ) guarantees that \Theta (E) exists in the sense of distributions. As in the C 1 case, the form i ss \Theta (E) is globally defined on X and independent of the choice of trivializations, and its de Rham cohomology class is the image of the first Chern class c1(E) 2 H 2 (X; Z) in H 2 DR (X; R). Before going further, we discuss two fundamental examples. 11.21. Example. Let D = P ffjDj be a divisor with coefficients ffj 2 Z and let E = O(D) be the associated invertible sheaf, defined as the sheaf of meromorphic functions u such that div(u) + D * 0. The corresponding line bundle can be given a singular metric defined by jjujj = juj (modulus of the meromorphic function u). If gj is a generator of the ideal of Dj on an open set \Omega ae X, then o/ (u) = u Q g ffj j defines a trivialization of O(D) on \Omega , thus our singular metric is associated to the weight ' = P ffj log jgjj. The Lelong-Poincar'e equation implies that i ss \Theta (O(D)) = dd c ' = [D]; where [D] = P ffj[Dj] denotes the current of integration on D. \Lambda 11.22. Example. Suppose that oe1; : : : ; oeN are non-zero holomorphic sections of E. One can then define a natural (possibly singular) Hermitian metric on E \Lambda , by setting jj, \Lambda jj 2 = X 1^j^n j, \Lambda :oej (x)j 2 for , \Lambda 2 E \Lambda x : The dual metric of E is given by jj, \Lambda jj 2 = jo/ (,)j 2 jo/ (oe1(x))j 2 + \Delta \Delta \Delta + jo/ (oeN (x))j 2 with respect to any local trivialization o/ . The associated weight function is therefore given by '(x) = log( P 1^j^N jo/ (oej (x))j 2 ) 1=2 . In this case ' is a psh function, therefore i\Theta (E) is a closed positive current. Denote by \Sigma the linear system defined by oe1; : : : ; oeN and B\Sigma = "oe \Gamma 1 j (0) its base locus. One has a meromorphic map \Phi \Sigma : XnB\Sigma ! P N\Gamma 1 ; x 7! [oe1(x) : oe2(x) : \Delta \Delta \Delta : oeN (x)]: With this notation, the curvature i 2ss \Theta (E) restricted to XnB\Sigma is identified with the inverse image by \Phi \Sigma of the Fubini-Study metric !FS = i 2ss d 0 d 00 log(jz1j 2 +\Delta \Delta \Delta +jzNj 2 ) on P N\Gamma 1 . It is therefore semi-positive. \Lambda 57 12. hodge theory of complete k"ahler manifolds 11.23. Ample and very ample line bundles. A holomorphic line bundle E on a compact complex manifold X is called a) very ample if the map \Phi jEj : X ! P N\Gamma 1 associated to the complete linear system jEj = P(H 0 (X; E)) is a regular embedding. (This implies in particular that the base locus is empty, i.e. BjEj = ;.) b) ample if there exists a multiple mE; m ? 0, which is very ample. We adopt here the additive notation for Pic(X) = H 1 (X; O \Lambda ), the symbol mE representing the line bundle E \Omega m . By refering to example 11.22, it follows that any ample line bundle E has a C 1 Hermitian metric, having a positive definite curvature form. Indeed, if the linear system jmEj gives an embedding in projective space, then one obtains a C 1 Hermitian metric on E \Omega m , and the m-th root gives a metric on E such that i 2ss \Theta (E) = 1 m \Phi \Lambda jmEj !FS. Conversely, Kodaira's embedding theorem [Kod54] says that any positive line bundle E is ample (see exercise 15.11 for a direct analytic proof of this fundamental theorem). 12. Hodge theory of complete K"ahler manifolds The goal of this section is primarily to extend to the case of complete K"ahler manifolds the results of Hodge theory already proven in the compact case. 12.A. Complete Riemannian manifolds. Before treating the complex situation, we will need to discuss some general results on the Hodge theory of complete Riemannian manifolds. Recall that a Riemannian manifold (M; g) is said to be complete if the geodesic distance ffig is complete, or what amounts to the same thing (Hopf-Rinow Lemma below), if the closed geodesic balls are all compact. We will need the following more precise characterization. 12.1. Lemma (Hopf-Rinow). The following properties are equivalent: a) (M; g) is complete; b) the closed geodesic balls Bg(a; r) are compact; c) there exists an exhaustive function 2 C 1 (M; R) such that jdjg ^ 1; d) there exists in M an exhaustive sequence (K*)*2N of compact sets and functions `* 2 C 1 (M; R) such that `* = 1 on a neighbourhood of K*; Supp `* ae K ffi *+1 ; 0 ^ `v ^ 1 and jd`*jg ^ 2 \Gamma * : Proof. a) =) b). The point x being fixed, one denotes by r0 = r0(x), the supremum of the real numbers r ? 0 such that Bg(a; r) is compact. Suppose r0 ! +1. Being given a sequence of points (x*) in Bg(a; r0) and ffl ? 0, one chooses a sequence of points x*;ffl 2 B(a; r0 \Gamma ffl) such that ffig(x* ; x*;ffl) ! 2ffl. By compactness of Bg(a; r0 \Gamma ffl), one can extract from (x*;ffl) a convergent subsequence for each ffl ? 0. By applying a diagonal process, one easily sees that one can extract from (x* ) a Cauchy subsequence. Consequently this sequence converges and Bg(a; r0) is compact. The local compactness of M implies that Bg(a; r0 + j) is still compact for j ? 0 small enough, which is a contradiction if r0 ! +1. b) =) c). Suppose M is connected. Choose a point x0 2 M and set 0(x) = 1 2 ffi(x0; x). Then 0 is exhaustive, and this is a Lipschitz function of order 1 2 , therefore 0 is differentiable almost everywhere on M . One obtains the sought for function by regularization. j.-p. demailly, part ii: L 2 estimations and vanishing theorems 58 c) =) d). Let be as in a) and let ae 2 C 1 (R; R) be a function such that ae = 1 on ] \Gamma 1; 1:1], ae = 0 on [1:99; +1[ and 0 ^ ae 0 ^ 2 on [1; 2]. Then K* = fx 2 M ; (x) ^ 2 *+1 g; `*(x) = ae \Gamma 2 \Gamma *\Gamma 1 (x) \Delta satisfies the desired properties. d) =) c). Set = P 2 *\Gamma 1 (1 \Gamma `*). c) =) b). The inequality jdjg ^ 1 implies j(x) \Gamma (y)j ^ ffig(x; y) for any x; y 2 M , therefore the geodesic ball Bg(a; r) ae fx 2 M ; ffig(x; a) ^ (a) + rg is relatively compact. b) =) a). This is obvious! \Lambda Let (M; g) be a Riemannian manifold, not necessarily complete for the moment, E a Hermitian vector bundle on M , with a given Hermitian connection D. One considers the unbounded operator between Hilbert spaces, still denoted by D D : L 2 (M; \Lambda p T \Lambda M \Omega E) ! L 2 (M; \Lambda p+1 T \Lambda M \Omega E); for which the domain Dom D is defined as follows: A section u 2 L 2 is said to be in Dom D if Du calculated in the sense of distributions is still in L 2 . The domain thus defined is always dense in L 2 , because Dom D contains the space D(M; \Lambda p T \Lambda M E) of C 1 sections with compact support, which is itself dense in L 2 . Moreover, the operator D thus defined, albeit not bounded, is closed, that is to say its graph is closed; this follows at once from the fact that the differential operators are continuous in the weak distribution topology. In the same way, the formal adjoint D \Lambda admits an extension to a closed operator D \Lambda : L 2 (M; \Lambda p+1 T \Lambda M \Omega E) ! L 2 (M; \Lambda p ; T \Lambda M \Omega E): Some well-known elementary results of spectral theory due to Von Neumann guarantees, in addition, the existence of a closed operator D \Lambda H with dense domain, called the Hilbert space adjoint of D, defined as follows: An element v 2 L 2 (M; \Lambda p+1 T \Lambda M \Omega E) is in Dom D \Lambda H if the linear form L 2 ! C ; u 7! hhDu; vii is continuous. It is thus written u 7! hhu; wii for a unique element w 2 L 2 (M; \Lambda p ; T \Lambda M \Omega E). One sets D \Lambda H v = w, so that D \Lambda H is defined by the usual adjoint relation hhDu; vii = hhu; D \Lambda H vii 8u 2 Dom D: (Note that the formal adjoint D \Lambda , itself, is defined by requiring only the validity of their relation for u 2 D(M; \Lambda p ; T \Lambda M \Omega E).) It is clear that one always has Dom D \Lambda H ae Dom D \Lambda and that D \Lambda H = D \Lambda on Dom D \Lambda H . In general, however, the domains are distinct (this is the case for example if M =]0; 1[; g = dx 2 ; D = d=dx !). A fundamental observation is that this phenomenon cannot occur if the Riemannian metric is complete. 12.2. Proposition. If the manifold (M; g) is complete, then: a) The space D(M; \Lambda ffl T \Lambda M E) is dense in Dom D, Dom D \Lambda and Dom D " Dom D \Lambda respectively, for the norms of the graphs u 7! jjujj + jjDujj; u 7! jjujj + jjD \Lambda ujj; u 7! jjujj + jjDujj + jjD \Lambda ujj: b) D \Lambda H = D \Lambda (i.e. the two domains coincide), and D \Lambda \Lambda H = D \Lambda \Lambda = D. 59 12. hodge theory of complete k"ahler manifolds c) Let \Delta = DD \Lambda + D \Lambda D be the Laplacian calculated in the sense of distributions. For any u 2 Dom \Delta ae L 2 (M; \Lambda ffl T \Lambda M \Omega E), one has hu; \Delta ui = jjDujj 2 +jjD \Lambda ujj 2 . In particular Dom \Delta ae Dom D " Dom D \Lambda ; Ker \Delta = Ker D " Ker D \Lambda ; and \Delta is self adjoint. d) If D 2 = 0, there is an orthogonal decomposition L 2 (M; \Lambda ffl T \Lambda M \Omega E) = H ffl L2 (M; E) \Phi Im D \Phi Im D \Lambda ; Ker D = H ffl L2 (M; E) \Phi Im D; where H ffl L2 (M; E) = fu 2 L 2 (M; \Lambda ffl T \Lambda M \Omega E); \Delta u = 0g is the space of L 2 harmonic forms on M . Proof. a) It is necessary to show for example that any element u 2 Dom D can be approximated in the norm of the graph of D by C 1 forms with compact support. By assumption, u and Du are in L 2 . Let (`* ) be a sequence of truncating functions as in Lemma 12.1 d). Then `*u ! u in L 2 (M; \Lambda ffl T \Lambda M \Omega E) and D(`*u) = `*Du + d`* ^ u where jd`* ^ uj ^ jd`*jjuj ^ 2 \Gamma * juj: Consequently d`* ^ u ! 0 and D(`* u) ! Du. By replacing u by `*u, one can assume that u has compact support, and with the aid of a partition of unity, one is reduced to the case where Supp u is contained in a coordinate chart of M on which E is trivial. Let (aeffl) be a family of regular kernels. A classical lemma in the theory of PDE (Friedrich's Lemma), shows that for any differential operator P of order 1 with C 1 coefficients, one has jjP (aeffl ? u) \Gamma aefflP ujjL2 ! 0, as ffl tends to 0 (u being an L 2 section with compact support in the coordinate chart considered). By applying this lemma to P = D; P = D \Lambda respectively, one arrives at the desired properties of density. b) is equivalent to the fact that hhDu; vii = hhu; D \Lambda vii; 8u 2 Dom D; 8v 2 Dom D \Lambda : However, according to a), one can find u*; v* 2 D(M; \Lambda ffl T \Lambda M \Omega E) such that u* ! u; v* ! v; Du* ! Du and D \Lambda v* ! D \Lambda v in L 2 (M; \Lambda ffl T \Lambda M \Omega E): The desired equality is then the limit of the equality hhDu*; v* ii = hhu* ; D \Lambda v*ii. c) Let u 2 Dom \Delta . Since \Delta u 2 L 2 and that \Delta is an elliptic operator of order 2, one obtains u 2 W 2 loc by applying the local version of the G*arding inequality. In particular Du; D \Lambda u 2 W 1 loc ae L 2 loc , and we can apply integration by parts as needed, after multiplying the respective forms by C 1 functions `* with compact support. Some simple calculations then give jj`*Dujj 2 + jj`*D \Lambda ujj 2 = = hh` 2 * Du; Duii + hhu; D(` 2 * D \Lambda u)ii = hhD(` 2 * ; u); Duii + hhu; ` 2 * DD \Lambda uii \Gamma 2hh`*d`* ^ u; Duii + 2hhu; `*d`* ^ D \Lambda uii = hh` 2 * u; \Delta uii \Gamma 2hhd`* ^ u; `*Duii + 2hhu; d`* ^ (`*D \Lambda u )ii ^ hh` 2 * u; \Delta uii + 2 \Gamma * \Gamma 2jj`*Dujjjjujj + 2jj`*D \Lambda ujjjjujj \Delta ^ hh` 2 * u; \Delta uii + 2 \Gamma * \Gamma jj`*Dujj 2 + jj`*D \Lambda ujj 2 + 2jjujj 2 \Delta : j.-p. demailly, part ii: L 2 estimations and vanishing theorems 60 Consequently jj`*Dujj 2 + jj`*D \Lambda ujj 2 ^ 1 1 \Gamma 2 \Gamma * \Gamma hh` 2 * u; \Delta uii + 2 1\Gamma * jjujj 2 \Delta : By letting * tend to +1, one obtains jjDujj 2 + jjD \Lambda ujj 2 ^ hhu; \Delta uii, in particular Du; D \Lambda u are in L 2 . This implies hhu; \Delta vii = hhDu; Dvii + hhD \Lambda u; D \Lambda vii; 8u; v 2 Dom \Delta ; because the equality holds for `*u and v, and that `* u ! u; D(`*u) ! Du and D \Lambda (`*u) ! D \Lambda u in L 2 . It follows from this that \Delta is self-adjoint. d) If P is a closed operator with dense domain on a Hilbert space H, then Ker P is closed and Ker P \Lambda = (Im P ) ? . Consequently (Ker P \Lambda ) ? = (Im P ) ?? = Im P . Since Ker P \Lambda itself is also closed, we have H = Ker P \Lambda \Phi (Ker P \Lambda ) ? = Ker P \Lambda \Omega Im P : This result applied to P = \Delta gives H ffl L2 (M; E) = Ker \Delta \Phi Im \Delta ; and it is clear according to (12.2 c) that Im \Delta ae Im D \Phi Im D \Lambda . Furthermore, one easily sees that Ker \Delta ; Im D and Im D \Lambda are pairwise orthogonal by using (12.2 a,c). Property d) follows as in the case where M is compact. \Lambda 12.3. Definition. Assume given a Riemannian manifold (M; g) and a Hermitian bundle E with a flat Hermitian connection D. We denote by H p DR;L2 (M; E), the L 2 de Rham cohomology groups, namely the cohomology groups of the complex (K ffl ; D) defined by K p = \Phi u 2 L 2 (M; \Lambda p T \Lambda M \Omega E); Du 2 L 2 \Psi : In other words, one has H p DR;L2 (M; E) = Ker D=Im D, where D is the L 2 extension of the connection calculated in the sense of distributions. Since H p L2 (M; E) = Ker D=Im D according to (12.2 d), it follows that: 12.4. Proposition. There is a canonical isomorphism H p L2 (M; E) ' H p DR;L2 (M; E)sep between H p L2 (M; E) and the separated space associated to the L 2 de Rham cohomology. In general the space H p DR;L2 (M; E) is not always separated, but it is in the important case where the L 2 cohomology is finite dimensional: 12.5. Corollary. If (M; g) is complete and if H p DR;L2 (M; E) is finite dimensional, then this space is separated and there is a canonical isomorphism H p L2 (M; E) ' H p DR;L2 (M; E): 61 12. hodge theory of complete k"ahler manifolds Proof. The space K p can be considered as the Hilbert space with norm u 7! (jjujjL2 + jjDujjL2) 1=2 . It is a question of seeing that Im D = D(K p\Gamma 1 ) is closed in Ker D, Ker D being itself closed in K p . Now D : K p\Gamma 1 ! Ker D is continuous and its image is of finite codimension by hypothesis. The fact that the image is closed is then a direct consequence of the Banach Theorem. \Lambda 12.6. Remark. For L 2 de Rham cohomology, observe that one obtains the identical cohomology groups when working with the subcomplex of global L 2 C 1 - forms, that is ~ K p = \Phi u 2 C 1 (M; \Lambda p T \Lambda M \Omega E); u 2 L 2 and Du 2 L 2 \Psi ae K p : For that, it suffices to construct an operator ~ K ffl ! K ffl which is a homotopic inverse to the inclusion. This can be done by using a regularization process by flows of vector fields tending to 0 sufficiently quickly, near infinity. 12.B. Case of Hermitian and complete K"ahler manifolds. The preceding results admit of course complex analogs, with almost identical proofs (the details will be therefore left to the reader). One says that a Hermitian or K"ahler manifold (X; !) is complete if the underlying Riemannian manifold is complete. 12.7. Proposition. Let (X; !) be a complete Hermitian manifold and E a Hermitian holomorphic vector bundle over X. There is a canonical isomorphism H p;q L2 (M; E) ' H p;q L2 (M; E)sep between the space of L 2 harmonic forms and the separated L 2 Dolbeault cohomology group, this latter space being itself equal to H p;q L2 (M; E) if the Dolbeault cohomology is finite dimensional. 12.8. Corollary. Let (X; !) be a K"ahler manifold and E a flat Hermitian bundle over X. a) Without further assumptions, there is, for any k, an orthogonal decomposition H k L2 (M; E) = M p+q=k H p;q L2 (M; E); H p;q L2 (M; E) = H q;p L2 (M; E \Lambda ): b) If moreover (X; !) is complete, there are canonical isomorphisms H k L2 (M; E)sep ' M p+q=k H p;q L2 (M; E)sep; H p;q L2 (M; E)sep ' H q;p L2 (M; E \Lambda )sep: c) If (X; !) is complete, and if the L 2 de Rham and Dolbeault cohomology groups are finite dimensional, there are canonical isomorphisms H k L2 (M; E) ' M p+q=k H p;q L2 (M; E); H p;q L2 (M; E) ' H q;p L2 (M; E \Lambda ): 12.C. Hodge theory of weakly pseudoconvex K"ahler manifolds. The weakly pseudoconvex K"ahler manifolds furnish an important example of complete K"ahler manifolds. j.-p. demailly, part ii: L 2 estimations and vanishing theorems 62 12.9. Definition. A complex manifold X is said to be weakly pseudoconvex if there exists a C 1 psh exhaustion function on X. (Recall that a function is said to be exhaustive if for any c ? 0 the level set Xc = \Gamma 1 (c) is relatively compact, i.e. (z) tends to +1 when z tends towards infinity, according to the stratification of the complements of compact parts of X.) In particular, the compact complete manifolds X are weakly pseudoconvex (take = 0), as well as the Stein manifolds. For example the affine algebraic subvarieties of C N (take (z) = jzj 2 ), the open balls X = B(z0; r) \Gamma take (z) = 1=(r \Gamma jz \Gamma z0j 2 ) \Delta , the open convex sets, and so on. A basic observation is the following: 12.10. Proposition. Any weakly pseudoconvex K"ahler manifold (X; !) has a complete K"ahler metric ^!. Proof. For any increasing convex function O/ 2 C 1 (R; R), we will consider the closed (1; 1)-form !O/ = ! + i d 0 d 00 (O/ ffi ) = ! + O/ 0 ()i d 0 d 00 + O/ 00 ()i d 0 ^ d 00 : Since the three terms are positive or zero, this is a K"ahler metric. The presence of the third term implies that the norm of O/ 00 () 1=2 d by comparison to !O/ is less than or equal to 1, therefore if ae is a choice of (O/ 00 ) 1=2 we have jd(ae ffi )j! O/ ^ 1. According to (12.1 c), !O/ will be complete as long as ae ffi is exhaustive, that is, as long as lim+1 ae(t) = +1. We therefore obtain the sufficient condition Z +1 t0 O/ 00 (t) 1=2 dt = +1; which is realized, for example, for the choice O/(t) = t 2 or O/(t) = t \Gamma log t; t * 1. \Lambda We have now established a Hodge decomposition theorem for weakly pseudoconvex K"ahler manifolds having "sufficiently many strictly pseudoconvex directions". Following Andreotti-Grauert [AG62], we introduce the: 12.11. Definition. A complex manifold X is said to be `-convex (resp. absolutely `-convex) if X has an exhaustion function (resp. a psh exhaustion function) , which is strongly `-convex on the complement XnK of a compact part, i.e. such that i d 0 d 00 has at least n \Gamma ` + 1 positive eigenvalues at any point of XnK, where n = dimC X. 12.12. Example. Let X be a smooth projective variety such that there exists a surjective morphism F : X ! Y onto another smooth projective variety Y . Let D be a divisor of Y and let X = XnF \Gamma 1 (D); Y = Y nD. We assume that F induces a submersion XnF \Gamma 1 (D) ! Y nD and that O(D)_D is ample. Then X is absolutely `-convex for ` = dim X \Gamma dim Y +1. Indeed, the hypothesis of ampleness of O(D)_D implies that there exists a Hermitian metric on O(D) for which the curvature is positive definite in a neighbourhood of D, that is on an open set of the form Y nK 0 where K 0 is a compact part of Y nD. Let oe 2 H 0 (Y ; O(D)) be the canonical section of the divisor D. Then \Gamma log joej 2 is strongly psh on Y nK 0 , consequently = \Gamma log joe ffi F j 2 is psh and strongly `-convex on XnK, where K = F \Gamma 1 (K 0 ). In addition, clearly defines an exhaustion on X. Nothing is known of on K, but 63 12. hodge theory of complete k"ahler manifolds it is enough to truncate by taking a maximal regularized C = maxffl(; C) with a constant C ? sup K to obtain an everywhere psh function C on X. We now can state the Hodge decomposition theorem for absolutely `-convex manifolds. This result is due to T. Ohsawa [Ohs81, 87]; we present here a simplified description of a proof of it in [Dem90a]. A purely algebraic approach of these results was obtained by Bauer-Kosarew [BaKo89,91] and [Kos91]. 12.13. Theorem (Ohsawa [Ohs81,87], [OT88]). Let (X; !) be a K"ahler manifold and n = dimC X, and assume that X is absolutely `-convex. Then, in suitable degrees, there is a Hodge decomposition and symmetry: H k DR (X; C ) ' M p+q=k H p;q (X; C ); H p;q (X; C ) ' H q;p (X; C ); k * n + `; H k DR;c (X; C ) ' M p+q=k H p;q c (X; C ); H p;q c (X; C ) ' H q;p c (X; C ); k ^ n \Gamma `; all these groups being finite dimensional (H k DR;c (X; C ) and H p;q c (X; C ) denotes here the cohomology groups with compact support). Moreover, there is a Lefschetz isomorphism ! n\Gamma p\Gamma q ^ ffl : H p;q c (X; C ) ! H n\Gamma q;n\Gamma p (X; C ); p + q ^ n \Gamma `: Proof. The finiteness of the de Rham cohomology groups concerned is easily obtained by means of Morse theory. Recall briefly the argument: a suitably small perturbation of a strongly `-convex exhaustion function gives a Morse function which is still strongly `-convex on the complement XnK of a compact set. The real Hessian D 2 of at a critical point induces a Hermitian form on the complexified tangent space C \Omega TX, and its restriction to T 1;0 X is identified with the complex Hessian i d 0 d 00 . Since the complex Hessian has by assumption at least n \Gamma ` + 1 positive eigenvalues on XnK, it follows from this that D 2 has at most 2n \Gamma (n \Gamma ` + 1) = n + ` \Gamma 1 negative eigenvalues on XnK, without which the positive and negative eigenvalues of D 2 would have a non-trivial intersection. Consequently all the critical points of index * n + ` are located in K and their number is finite. This implies that the groups H k DR (X; C ) of degree k * n + ` are finite dimensional. The finiteness of the Dolbeault cohomology groups H p;q (X; C ) = H q (X; \Omega p X ) is a result of the theorem of Andreotti-Grauert [AG62] (all the cohomology groups of higher degree than ` with values in a given coherent sheaf are separated and finite dimensional if the manifold is `-convex). It is noted however, that the `- convexity, although sufficient to ensure the finiteness of the various groups involved, is not sufficient to guarantee the existence of a Hodge decomposition, nor even the Hodge symmetry. The reader will find a simple counterexample in GrauertRiemenschneider [GR70]. Now let ! be a K"ahler metric on X and a strongly `-convex psh exhaustion function on XnK. As one can see, the existence of a Hodge decomposition follows directly from the fact that one has such a decomposition for the L 2 harmonic forms. The key point resides in the observation that any L 2 loc form of degree k * n + ` becomes globally L 2 for a suitable choice of metric !O/ = ! + i d 0 d 00 (O/ ffi ). The groups H k DR (X; C ) and H p;q (X; C ) could then be considered as the inductive limit of L 2 cohomology groups. In the sequel, we will use notation such as j.-p. demailly, part ii: L 2 estimations and vanishing theorems 64 L 2 !O/ (X; \Lambda p;q T \Lambda X ); H p;q L2;!O/ (X; C ), to denote the spaces of L 2 -forms (resp. harmonic forms) relative to !O/. Since !O/ is K"ahler, one has (12.14) H k L2;!O/ (M; C ) = M p+q=k H p;q L2;!O/ (M; C ); H p;q L2;!O/ (M; C ) = H q;p L2;!O/ (M; C ); with an isomorphism H k L2;!O/ (M; C ) ' H k L2;!O/ (M; C )sep as long as !O/ is complete. In the sequel, we always assume that !O/ is complete. It is enough, for example, to impose O/ 00 (t) * 1 on [0; +1[. 12.15. Lemma. Let u be a form of bidegree (p; q) with L 2 loc coefficients on X. If p + q * n + `, then u 2 L 2 !O/ (X; \Lambda p;q T \Lambda X ) as long as O/ grows sufficiently quickly near infinity. Proof. At a fixed point x 2 X, there exists an orthogonal basis (@=@z1; : : : ; @=@zn) of TX;x for which !(x) = i X 1^j^n dzj ^ dzj; !O/(x) = i X 1^j^n *j (x)dzj ^ dzj; where *1 ^ \Delta \Delta \Delta ^ *n are the eigenvalues of !O/ relative to !. Then the volume elements dV! = ! n =2 n n! and dV! O/ = ! n O/ =2 n n! are bound by the relation dV! O/ = *1 \Delta \Delta \Delta *ndV!; and for a (p; q)-form u = P I;J uI;JdzI ^ dzJ we find that juj 2 !O/ = X jIj=p;jJj=q ` Y k2I *k Y k2J *k '\Gamma 1 juI;Jj 2 : In particular, it follows that juj 2 !O/ dV! O/ ^ *1 \Delta \Delta \Delta *n *1 \Delta \Delta \Delta *p*1 \Delta \Delta \Delta *q juj 2 ! dV! = *p+1 \Delta \Delta \Delta *n *1 \Delta \Delta \Delta *q juj 2 ! dV!: In addition, one has upper bounds *j ^ 1 + C1O/ 0 (); 1 ^ j ^ n \Gamma 1; *n ^ 1 + C1O/ 0 () + C2O/ 00 () where C1(x) is the largest eigenvalue of i d 0 d 00 (x) and C2(x) = j@(x)j 2 . For to obtain the first n \Gamma 1 inequalities, one need only apply the minimum principal on the kernel of @. Since i d 0 d 00 has at most ` \Gamma 1 zero eigenvalues on XnK, the minimum principal also gives lower bounds *j * 1; 1 ^ j ^ ` \Gamma 1; *j * 1 + cO/ 0 (); ` ^ j ^ n; where c(x) * 0 is the `-th eigenvalue of i d 0 d 00 (x) and c(x) ? 0 on XnK. If we assume O/ 0 * 1, then we can easily deduce juj 2 !O/ dV! O/ juj 2 ! dV! ^ \Gamma 1 + C1O/ 0 ()) n\Gamma p\Gamma 1 (1 + C1O/ 0 () + C2O/ 00 () \Delta \Gamma 1 + cO/ 0 () \Delta q\Gamma `+1 ^ C3 \Gamma O/ 0 () n+`\Gamma p\Gamma q\Gamma 1 + O/ 00 ()O/ 0 () \Gamma 2 \Delta on XnK: For p + q * n + `, this is smaller or equal to C3 \Gamma O/ 0 () \Gamma 1 + O/ 00 ()O/ 0 () \Gamma 2 \Delta ; 65 12. hodge theory of complete k"ahler manifolds and it is easy to show that this quantity can be made arbitrarily small towards infinity on X as O/ grows sufficiently quickly to infinity on R. \Lambda Proof of the theorem (12.13), conclusion. A well-known result of the Andreotti-Grauert [AG62] guarantees that the natural topology of the cohomology groups H q (X; F) of any given coherent sheaf F on a `-convex manifold is separated for q * `. If F = O(E) is the sheaf of sections of a holomorphic vector bundle, the groups H q (X; O(E)) are algebraically and topologically isomorphic to the cohomology groups of the Dolbeault complex of forms of type (0; q) with L 2 loc coefficients for which the d 00 -differential has L 2 coefficients in terms of the Fr'echet topology defined by the semi-norms u 7! jjujjL2(K) + jjd 00 ujjL2(K). To see this, one can begin again word for word the proof of Theorem (1.3), by observing that the L 2 loc complex still furnishes a resolution of O(E) by the (acyclic) sheaves of C 1 -modules. It follows from what proceeds this that the morphism L 2 !O/ (X; \Lambda p;q T \Lambda X ) oe Ker D 00 !O/ ! H p;q (X; C ) = H q (X; \Omega p X ) is continuous and with closed kernel. Consequently this kernel contains the image Im D 00 !O/ , and we obtain a factorization H p;q !O/ (X; C ) ' Ker D 00 !O/ =Im D 00 !O/ ! H p;q (X; C ): The proof of proposition (12.2) further shows that Im D 00 !O/ coincides with the image of D 00 \Gamma D(X; \Lambda p;q T \Lambda X ) \Delta in L 2 !O/ (X; \Lambda p;q T \Lambda X ). Consider the limit morphism (12.16) lim \Gamma ! O/ H p;q !O/ (X; C ) ! H p;q (X; C ); where the inductive limit is extended to the set of increasing C 1 convex functions O/, such that O/ 00 (t) * 1 on [0; +1[, with the order relation O/1 _ O/2 () O/1 ^ O/2 and L 2 !O/ 1 (X; \Lambda p;q T \Lambda X ) ae L 2 !O/ 2 (X; \Lambda p;q T \Lambda X ) for k = p + q: It is easy to see that this order is filtered by again taking the arguments used for Lemma (12.15). Furthermore, it is well-known that the de Rham cohomology groups are always separated in the induced topology from the Fr'echet topology on the space of forms, consequently one has a limit morphism (12.16DR) lim \Gamma ! O/ H k !O/ (X; C ) ! H k DR (X; C ) analogous to (12.16). The decomposition formula of Theorem (12.13) follows now from (12.14), and from the following elementary lemma. 12.17. Lemma. The limit morphisms (12:16); (12:16)DR are bijective for k = p + q * n + `. Proof. Let us treat for example the case of the morphism (12.16), and let u be a L 2 loc d 00 -closed form of bidegree (p; q); p + q * n + `. Then there exists a choice of O/ for which u 2 L 2 !O/ , therefore u 2 Ker D 00 !O/ and (12.16) is surjective. If a class fug 2 H p;q !O/ 0 (X; C ) is sent to zero in H p;q (X; C ), one can write u = d 00 v for a certain form v with L 2 loc coefficients and of bidegree (p; q \Gamma 1). In the case p + q ? n + `, we will have v 2 L 2 !O/ for O/ * O/0 large enough, therefore the class of u = D 00 !O/ v in H p;q !O/ (X; C ) is zero and (12.16) is injective. When p + q = n + `, the form v does j.-p. demailly, part ii: L 2 estimations and vanishing theorems 66 not necessarily belong anymore to one of the spaces L 2 !O/ , but it suffices to show that u = d 00 v is in the image of Im D 00 !O/ for O/ large enough. Let ` 2 C 1 (R; R) be a truncating function such that `(t) = 1 for t ^ 1=2; `(t) = 0 for t * 1 and j` 0 j ^ 3. Then d 00 (`(ffl)v) = `(ffl)d 00 v + ffl` 0 (ffl)d 00 ^ v: According to the proof of lemma (12.15), there exists a continuous function C(x) ? 0 such that jvj 2 !O/ dV! O/ ^ C \Gamma 1 + O/ 00 ()=O/ 0 () \Delta jvj 2 ! dV!, whereas jd 00 j 2 !O/ ^ 1=O/ 00 () according to the same definition of !O/. We see therefore that the integral Z X j` 0 (ffl)d 00 ^ vj 2 !O/ dV! O/ ^ Z X C \Gamma 1=O/ 00 () + 1=O/ 0 () \Delta jvj 2 dV is finite for O/ large enough, and by dominated convergence d 00 (`(ffl)v) converges to d 00 v = u in L 2 !O/ (X; \Lambda p;q T \Lambda X ). \Lambda Poincar'e and Serre duality show that the spaces H k DR;c (X; C ) and H p;q c (X; C ) with compact support are dual to the spaces H 2n\Gamma k DR (X; C ) and H n\Gamma p;n\Gamma q (X; C ) since the latter are separated and of finite dimension, which is very much the case if k = p + q ^ n \Gamma `. We therefore obtain a dual Hodge decomposition (12.18) H k c (X; C ) ' M p+q=k H p;q c (X; C ); H p;q c (X; C ) ' H q;p c (X; C ); k ^ n \Gamma `: In addition, it is easy to prove that the Lefschetz isomorphism (12.19) ! n\Gamma p\Gamma q O/ ^ ffl : H p;q !O/ (X; C ) ! H n\Gamma q;n\Gamma p !O/ (X; C ) given in the limit is an isomorphism between the cohomology with compact support and cohomology without supports (this result is due to Ohsawa [Ohs81]). Indeed, if p + q ^ n \Gamma `, the natural morphism (12.20) H p;q c (X; C ) = Ker D 00 D =Im D 00 D ! Ker D 00 !O/ =Im D 00 !O/ ' H p;q !O/ (X) is dual to the morphism H n\Gamma p;n\Gamma q !O/ (X; C ) ! H n\Gamma p;n\Gamma q (X; C ), which is surjective for O/ large enough according to Lemma (12.17) and the finiteness of the group H n\Gamma p;n\Gamma q (X; C ). Therefore (12.20) is injective for O/ large, and after composition with the Lefschetz isomorphism (12.19), we obtain an injection ! n\Gamma p\Gamma q ^ ffl = ! n\Gamma p\Gamma q O/ ^ ffl : H p;q c (X; C ) ! H n\Gamma q;n\Gamma p L2;!O/ (X; C )sep ' H n\Gamma q;n\Gamma p !O/ (X; C ): (The equality ! n\Gamma p\Gamma q ^ ffl = ! n\Gamma p\Gamma q O/ ^ ffl follows from the fact that !O/ has the same cohomology class as !.) By taking the inductive limit on O/ and in combination with the limit isomorphism (12.16), we obtain an injective map (12.21) ! n\Gamma p\Gamma q ^ ffl : H p;q c (X; C ) ! H n\Gamma q;n\Gamma p (X; C ); p + q ^ n \Gamma `: Since the two groups have the same dimension by the Serre duality theorem and Hodge symmetry, the map is necessarily an isomorphism. \Lambda 67 13. bochner techniques and vanishing theorems 12.22. Remark. Since the Lefschetz isomorphism (12.21) can be factored through H p;q (X; C ) or through H n\Gamma q;n\Gamma p c (X; C ), we deduce from this that the natural morphisms H p;q c (X; C ) ! H p;q (X; C ) are injective for p + q ^ n \Gamma ` and surjective for p + q * n + `. Of course, there are entirely analogous properties for the de Rham cohomology groups. 13. Bochner techniques and vanishing theorems Let X be a complex manifold with a given K"ahler metric ! = P !jkdzj ^ dzk. Let (E; h) be a Hermitian holomorphic vector bundle over X. We denote by D = D 0 + D 00 the Chern connection and \Theta (E) the associated curvature tensor. 13.1. Basic commutivity relations. Let L be the operator Lu = ! ^ u acting on the vector valued forms, and let \Lambda = L \Lambda be its adjoint. Then [D 00\Lambda ; L] = i d 0 ; [D 0\Lambda ; L] = \Gamma i d 00 ; [\Lambda ; D 00 ] = \Gamma i d 0\Lambda ; [\Lambda ; D 0 ] = i d 00\Lambda : Proof (outline). This is a simple consequence of the commutivity relation (6.14) already shown for the trivial connection d = d 0 + d 00 on E = X \Theta C . Indeed, for any point x0 2 X, there exists a local holomorphic frame (e*)1^*^r of E such that he*; e_i = ffi*_ + O(jzj 2 ): (The proof is identical to that of Theorem 5.8.) For s = P s* \Omega e* with s* 2 C 1 (X; \Lambda p;q T \Lambda X ), we obtain D 00 s = X d 00 s* \Omega e* + O(jzj); D 00\Lambda s = X d 00\Lambda s* \Omega e* + O(jzj): The stated relations follow easily. \Lambda 13.2. The Bochner-Kodaira-Nakano identity. If (X; !) is a K"ahler manifold, the complex Laplacians \Delta 0 and \Delta 00 acting on the forms with values in E satisfy the identity \Delta 00 = \Delta 0 + [i \Theta (E); \Lambda ]: Proof. The latter equality (13.1) gives D 00\Lambda = \Gamma i[\Lambda ; D 0 ], therefore \Delta 00 = [D 00 ; D 00\Lambda ] = \Gamma i \Theta D 00 ; [\Lambda ; D 0 ] \Lambda : The Jacobi identity implies \Theta D 00 ; [\Lambda ; D 0 ] \Lambda = \Theta \Lambda ; [D 0 ; D 00 ] \Lambda + \Theta D 0 ; [D 00 ; \Lambda ] \Lambda = [\Lambda ; \Theta (E)] + i[D 0 ; D 0\Lambda ]; which is based on the fact that [D 0 ; D 00 ] = D 2 = \Theta (E). The stated identity follows. \Lambda Assume that X is compact and let u 2 C 1 (X; \Lambda p;q T \Lambda X \Omega E) be an arbitrary (p; q)-form. Integration by parts gives h\Delta 0 u; ui = jjD 0 ujj 2 + jjD 0\Lambda ujj 2 * 0; j.-p. demailly, part ii: L 2 estimations and vanishing theorems 68 and one has an analogous equality for \Delta 00 . From the Bochner-Kodaira-Nakano identity, one deduces a priori the inequality (13.3) jjD 00 ujj 2 + jjD 00\Lambda ujj 2 * Z X h[i \Theta (E); \Lambda ]u; uidV!: This inequality is the well-known Bochner-Kodaira-Nakano inequality (see [Boc48], [Kod53], [Nak55]). When u is \Delta 00 -harmonic, we obtain Z X h[i \Theta (E); \Lambda ]u; uidV ^ 0: If the Hermitian operator [i \Theta (E); \Lambda ] is positive on each fiber of \Lambda p;q T \Lambda X \Omega E, then one sees that u is necessarily zero, therefore H p;q (X; E) = H p;q (X; E) = 0 according to Hodge theory. In this approach, the essential point is to know how to calculate the curvature form \Theta (E) and to find sufficient conditions for which the operator [i \Theta (E); \Lambda ] is positive definite. Some elementary (albeit somewhat agonizing) calculations yields the following formula: If the curvature of E is written in the form (11.16) and if u = X uJ;K;*dzI ^ dzJ \Omega e*; jJj = p; jKj = q; 1 ^ * ^ r is a (p; q)-form with values in E, then h[i \Theta (E); \Lambda ]u; ui = X j;k;*;_;J;S cjk*_uJ;jS;*uJ;kS;_(13.4) + X j;k;*;_;R;K cjk*;_ukR;K;*ujR;K_ \Gamma X j;*;_;J;K cjj*_uJ;K;*uJ;K;_; where the summations are extended to all the indices 1 ^ j; k ^ n; 1 ^ *; _ ^ r and all the multi-indices jJj = p; jKj = q; jRj = p \Gamma 1; jSj = q \Gamma 1. (Here the notation uJK* is applied to some not necessarily increasing multi-indices. Also, it is agreed that the sign of this coefficient is alternating, under the action of permutations.) Taking into account the complexity of the curvature term (13.4), the sign of this term is in general difficult to elucidate, except in some very particular case. The simpler case is the case p = n. All of the terms of the extra second summation in (13.4) are then such that j = k and R = f1; : : : ; ngnfjg. Consequently the second and third summations are equal. It follows that h[i \Theta (E); \Lambda ]u; ui = X j;k;*;_;J;S cjk*_uJ;jS;*uJ;kS;_ is positive on the (n; q)-forms under the hypothesis that E is positive in the sense of Nakano. In this case, X is automatically K"ahler since ! = TrE(i \Theta (E)) = i X j;k;* cjk**dzj ^ dzk = i \Theta (det E) therefore defines a K"ahler metric. 69 13. bochner techniques and vanishing theorems 13.5. Nakano Vanishing Theorem (1955). Let X be a compact complex manifold and let E be a positive vector bundle in the sense of Nakano on X. Then H n;q (X; E) = H q (X; KX \Omega E) = 0 for all q * 1: \Lambda Another approachable case is the case where E is a line bundle (r = 1). Indeed, at each point x 2 X, we can then choose a coordinate system, which simultaneously diagonalizes the Hermitian forms !(x) and \Theta (E)(x), in such a way that !(x) = i X 1^j^n dzj ^ dzj; \Theta (E)(x) = i X 1^j^n fljdzj ^ dzj with fl1 ^ \Delta \Delta \Delta ^ fln. The eigenvalues of curvature flj = flj(x) are then defined in a unique way and depend continuously in x. In the former notation, we have flj = cjj11 and all the other coefficients cjk*_ are zero. For any (p; q)-form u = P uJKdzJ ^ dzK \Omega e1, this gives h[i \Theta (E); \Lambda ]u; ui = X jJj=p;jKj=q ` X j2J flj + X j2K flj \Gamma X 1^j^n flj ' juJKj 2 * (fl1 + \Delta \Delta \Delta + flq \Gamma fln\Gamma p+1 \Gamma \Delta \Delta \Delta \Gamma fln)juj 2 :(13.6) Assume that i \Theta (E) is positive. It is then natural to provide X with the particular K"ahler metric ! = i \Theta (E). Then flj = 1 for j = 1; 2; : : : ; n and we obtain h[i \Theta (E); \Lambda ]u; ui = (p + q \Gamma n)juj 2 : As a consequence: 13.7. Kodaira-Akizuki-Nakano Vanishing Theorem ([AN54]). If E is a positive line bundle over a compact complex manifold X, then H p;q (X; E) = H q (X; \Omega p X \Omega E) = 0 for p + q * n + 1: \Lambda More generally, if E is a positive vector bundle in the sense of Griffiths (or ample), of rank r * 1, Le Potier [LP75] has proven that H p;q (X; E) = 0 for p + q * n + r. The proof is not a direct consequence of the Bochner technique. A simple enough proof has been obtained by M. Schneider [Sch74], by utilizing the Leray spectral sequence associated to the projection on X of the projective bundle P(E) ! X. 13.8. Exercise. It is significant for various applications to formulate vanishing theorems which are also valid in the case of semi-positive line bundles. There is, for example, the following result due to J. Girbau [Gir76] : Let (X; !) be a compact K"ahler manifold, assume that E is a line bundle and that i \Theta (E) * 0 has at least n \Gamma k positive eigenvalues at each point, for a certain integer k * 0. Then H p;q (X; E) = 0 for p + q * n + k + 1. Indication. Use the K"ahler metric !ffl = i \Theta (E) + ffl! with small ffl ? 0. A more natural and powerful version of this result has been obtained by A. Sommese [Som78, ShSo85] : Following these authors, we say that E is k-ample if a certain multiple mE is such that the canonical map \Phi jmEj : XnBjmEj ! P N\Gamma 1 j.-p. demailly, part ii: L 2 estimations and vanishing theorems 70 has all its fibers of dimension ^ k and dim BjmEj ^ k. If X is projective and if E is k-ample, then H p;q (X; E) = 0 for p + q * n + k + 1. Indication. Prove the dual result, that H p;q (X; E \Gamma 1 ) = 0 for p+q ^ n\Gamma k\Gamma 1, by induction on k. First show that E is 0-ample if and only if E is positive. Then use some hyperplane sections Y ae X to prove the induction step, by considering the exact sequences 0 ! \Omega p X \Omega E \Gamma 1 \Omega O(\Gamma Y ) ! \Omega p X \Omega E \Gamma 1 ! (\Omega p X \Omega E \Gamma 1 )_Y ! 0; 0 ! \Omega p\Gamma 1 Y \Omega E \Gamma 1 _Y ! (\Omega p X \Omega E \Gamma 1 )_Y ! \Omega p Y \Omega E \Gamma 1 _Y ! 0: \Lambda 14. L 2 estimations and existence theorems The starting point is the following L 2 existence theorem, which is essentially due to H"ormander [H"or65, 66], and Andreotti-Vesentini [AV65]. We only sketch the principal ideas, while referring for example to [Dem82] for a detailed exposition of the techniques considered in the situation here. 14.1. Theorem. Let (X; !) be a complete K"ahler manifold, and let E be a Hermitian vector bundle of rank r on X, such that the curvature operator A = A p;q E;! = [i \Theta (E); \Lambda !] is semi-positive on all the fibers of \Lambda p;q T \Lambda X \Omega E; q * 1. Let g 2 L 2 (X; \Lambda p;q T \Lambda X \Omega E) be a form satisfying D 00 g = 0 and Z X hA \Gamma 1 g; gidV! ! +1: (At the points where A is not positive definite, we assume as a precondition that A \Gamma 1 g exists almost everywhere. We then choose the preconditional term A \Gamma 1 g of minimal norm, orthogonal to Ker A.) Then there exists f 2 L 2 (X; \Lambda p;q\Gamma 1 T \Lambda X \Omega E) such that D 00 f = g and Z X jf j 2 dV! ^ Z X hA \Gamma 1 g; gidV!: Proof. Let u 2 L 2 (X; \Lambda p;q T \Lambda X \Omega E) be a form such that D 00 u 2 L 2 and D 00\Lambda u 2 L 2 in the sense of distributions. Lemma (12.2 a) shows (under the indispensable hypothesis that ! is complete) that u is the limit of a sequence of C 1 forms u* with compact support in such a way that u* ! u; D 00 u* ! D 00 u and D 00\Lambda u* ! D 00\Lambda u in L 2 . It follows that a priori the inequality (13.3) extends to arbitrary forms u such that u; D 00 u; D 00\Lambda u 2 L 2 . Now, since Ker D 00 is weakly (and therefore strongly) closed, we obtain an orthogonal decomposition of the Hilbert space L 2 (X; \Lambda p;q T \Lambda X \Omega E), namely L 2 (X; \Lambda p;q T \Lambda X \Omega E) = Ker D 00 \Phi (Ker D 00 ) ? : Let v = v1 + v2 be the corresponding decomposition of a C 1 form v 2 D p;q (X; E) with compact support (in general, v1; v2 do not have compact support!). Since (Ker D 00 ) ? = Im D 00\Lambda ae Ker D 00\Lambda by duality and g; v1 2 Ker D 00 by hypothesis, we obtain D 00\Lambda v2 = 0 and jhg; vij 2 = jhg; v1ij 2 ^ Z X hA \Gamma 1 g; gidV! Z X hAv1; v1idV! 71 14. L 2 estimations and existence theorems by applying the Cauchy-Schwartz inequality. The inequality (13.3) a priori, applied to u = v1 gives Z X hAv1; v1idV! ^ jjD 00 v1jj 2 + jjD 00\Lambda v1jj 2 = jjD 00\Lambda v1jj 2 = jjD 00\Lambda vjj 2 : Combining these two inequalities we find that jhg; vij 2 ^ `Z X hA \Gamma 1 g; gidV! ' jjD 00\Lambda vjj 2 for any C 1 (p; q)-form v with compact support. This shows that there is a welldefined linear form w = D 00\Lambda v 7! hv; gi; L 2 (X; \Lambda p;q\Gamma 1 T \Lambda X \Omega E) oe D 00\Lambda (D p;q (E)) ! C on the image of D 00\Lambda . This linear form is continuous in the L 2 norm, and its norm is ^ C with C = `Z X hA \Gamma 1 g; gidV! '1=2 : According to the Hahn-Banach Theorem, there exists an element f 2 L 2 (X; \Lambda p;q\Gamma 1 T \Lambda X \Omega E) such that jjf jj ^ C and hv; gi = hD 00\Lambda v; f i for any v, consequently D 00 f = g in the sense of distributions. The inequality jjf jj ^ C is equivalent to the latter estimation in the theorem. \Lambda The preceding L 2 existence theorem can be applied in the general context of weakly pseudoconvex K"ahler manifolds (see definition (12.9)), and the same if the K"ahler metric considered ! is not complete. Indeed, according to Proposition (12.10), we arrive at complete K"ahler metrics by setting !ffl = ! + ffli d 0 d 00 2 = ! + 2ffl(2i d 0 d 00 + i d 0 ^ d 00 ) with a C 1 psh exhaustion function * 0. As a consequence, the L 2 existence theorem (14.1) applies to each K"ahler metric !ffl. Indeed one can show (the calculations being left to the reader!) that the quantities jgj 2 ! dV! and h(A p;q E;! ) \Gamma 1 g; gi!dV! are decreasing functions of ! when p = n = dimC X. For a D 00 -closed form g of bidegree (n; q), we therefore obtains solutions fffl of the equation D 00 fffl = g satisfying Z X jffflj 2 !ffl dV! ffl ^ Z X h(A p;q E;!ffl ) \Gamma 1 g; gi! ffl dV! ffl ^ Z X h(A p;q E;! ) \Gamma 1 g; gi!dV!: These solutions fffl can be uniformly bounded in the L 2 norm on any compact set. Thus we can extract a weakly convergent subsequence in L 2 . The limit f is a solution of D 00 f = g and satisfies the required L 2 estimation relative to the metric ! initially given (which, to repeat, is not necessarily complete). A particularly important case is the following: j.-p. demailly, part ii: L 2 estimations and vanishing theorems 72 14.2. Theorem. Let (X; !) be a K"ahler manifold, dim X = n. Assume that X is weakly pseudoconvex. Let E be a Hermitian line bundle and let fl1(x) ^ \Delta \Delta \Delta ^ fln(x) be the eigenvalues of curvature (i.e. the eigenvalues of i \Theta (E) with respect to the metric !) at any point x. Assume that the curvature is semi-positive, i.e. fl1 * 0 everywhere. Then for any form g 2 L 2 (X; \Lambda n;q T \Lambda X \Omega E) satisfying D 00 g = 0 and Z X (fl1 + \Delta \Delta \Delta + flq) \Gamma 1 jgj 2 dV! ! +1; (one assumes therefore g(x) = 0 almost everywhere at all points where fl1(x) + \Delta \Delta \Delta + flq(x) = 0), there exists f 2 L 2 (X; \Lambda n;q\Gamma 1 T \Lambda X \Omega E) such that D 00 f = g and 2 Z X jf j 2 dV! ^ Z X (fl1 + \Delta \Delta \Delta + flq) \Gamma 1 jgj 2 dV!: Proof. Indeed, for p = n, formula (13.6) shows that hAu; ui * (fl1 + \Delta \Delta \Delta + flq)juj 2 ; therefore hA \Gamma 1 u; ui * (fl1 + \Delta \Delta \Delta + flq) \Gamma 1 juj 2 . \Lambda An important observation is that the above theorem still applies when the Hermitian metric of E is a singular metric with positive curvature in the sense of currents. Indeed, by a process of regularization (convolution of psh functions by regular kernels), the metric can made C 1 and the solutions obtained by means of Theorems (14.1) or (14.2), since the regular metrics have limits satisfying the desired estimates. In particular, we obtain the following corollary. 14.3. Corollary. Let (X; !) be a K"ahler manifold, dim X = n. Assume that X is weakly pseudoconvex. Let E be a holomorphic bundle provided with a singular metric for which the local weight is denoted by ' 2 L 1 loc . Assume that i \Theta (E) = 2i d 0 d 00 ' * ffl! for a certain ffl ? 0. Then for any form g 2 L 2 (X; \Lambda n;q T \Lambda X \Omega E) satisfying D 00 g = 0, there exists f 2 L 2 (X; \Lambda p;q\Gamma 1 T \Lambda X \Omega E) such that D 00 f = g and Z X jf j 2 e \Gamma 2' dV! ^ 1 qffl Z X jgj 2 e \Gamma 2' dV!: \Lambda We denoted here somewhat incorrectly the metric in the form jf j 2 e \Gamma 2' , as if the weight ' were globally defined on X (certainly, this is not possible if E is globally trivial). By abuse of notation, we will nevertheless use this same notation because it clearly underlines the dependence of the L 2 norm on the psh function associated to the weight. 15. Vanishing theorems of Nadel and Kawamata-Viehweg We begin by introducing the concept of multiplier ideal sheaves, following A. Nadel [Nad89]. The principal idea in fact goes back to the fundamental work of E. Bombieri [Bom70] and H. Skoda [Sko72]. 73 15. vanishing theorems of nadel and kawamata-viehweg 15.1. Definition. Let ' be a psh function on an open set \Omega ae X. We associate to ', the sheaf of ideals J (') ae O\Omega formed from the germs of holomorphic functions f 2 O\Omega ;x such that jf j 2 e \Gamma 2' is integrable with respect to the Lebesgue measure in the local coordinates x. This sheaf will be called the multiplier ideal sheaf associated to the weight '. The variety of zeros V (J (')) is therefore the set of points in a neighbourhood for which e \Gamma 2' is non-integrable. Of course, such points cannot appear where ' has logarithmic poles. The precise formulation is the following. 15.2. Definition. We say that a psh function ' has a logarithmic pole with coefficient fl at a point x 2 X if the Lelong number *('; x) := lim inf z!x '(z) log jz \Gamma xj is non-zero and if *('; x) = fl. 15.3. Lemma. (Skoda [Sko72]). Let ' be a psh function on an open set \Omega ae C n and let x 2 \Omega . a) If *('; x) ! 1, then e \Gamma 2' is integrable in a neighbourhood of x, in particular J (')x = O\Omega ;x. b) If *('; x) * n + s for a certain integer s * 0, then e \Gamma 2' * Cjz \Gamma xj \Gamma 2n\Gamma 2s in a neighbourhood of x and J (')x ae m s+1 \Omega ;x , where m\Omega ;x denotes the maximal ideal of O\Omega ;x. Proof. The proof rests on some classical estimations of complex potential theory, see H. Skoda [Sko72]. \Lambda 15.4. Proposition ([Nad89]). For any psh function ' on \Omega ae X, the sheaf J (') is a coherent sheaf of ideals on \Omega . Proof. Since the result is local we can assume that \Omega is the unit ball in C n . Let H'(\Omega ) be the set of the holomorphic functions f on \Omega such that R \Omega jf j 2 e \Gamma 2' d* ! +1. According to the strong Noetherian property of coherent sheaves, the set H'(\Omega ) generates a coherent sheaf of ideals J ae O\Omega . It is clear that J ae J ('); for to show equality, it suffices to verify that Jx + J (')x " m s+1 \Omega ;x = J (')x for any integer s, by virtue of Krull's lemma. Let f 2 J (')x be a germ defined on a neighbourhood V of x and let ` be a truncating function with support in V , such that ` = 1 in a neighbourhood of x. We can solve the equation d 00 u = g := d 00 (`f ) by means of L 2 estimations of H"ormander (14.3), where E is the trivial line bundle \Omega \Theta C provided with the strictly psh weight ~'(z) = '(z) + (n + s) log jz \Gamma xj + jzj 2 : We obtain a solution u such that R \Omega juj 2 e \Gamma 2' jz \Gamma xj \Gamma 2(n+s) d* ! 1, therefore F = `f \Gamma u is holomorphic, F 2 H'(\Omega ) and fx \Gamma Fx = ux 2 J (')x " m s+1 \Omega ;x . This proves our assertion. \Lambda The multiplier ideal sheaves satisfy the following essential functorial property, relative to the direct images of sheaves by modifications. j.-p. demailly, part ii: L 2 estimations and vanishing theorems 74 15.5. Proposition. Let u : X 0 ! X be a modification of non-singular complex varieties (i.e. a proper holomorphic map that is generically 1 : 1), and let ' be a psh function on X. Then _\Lambda \Gamma O(KX0) \Omega J (' ffi _) \Delta = O(KX ) \Omega J ('): Proof. Let n = dim X = dim X 0 and let S ae X be an analytic subvariety such that _ : X 0 nS 0 ! XnS is a biholomorphism. By definition of multiplier ideal sheaves, O(KX ) \Omega J (') is identified with the sheaf of holomorphic n-forms f on some open set U ae X, satisfying i n 2 f ^f e \Gamma 2' 2 L 1 loc (U ). Since ' is locally bounded above, we can likewise consider the forms f which a priori are defined only on U nS, because f is in L 2 loc (U ), and thus automatically extends through S. The change of variables formula gives Z U i n 2 f ^ f e \Gamma 2' = Z _\Gamma 1(U) i n 2 _ \Lambda f ^ _ \Lambda f e \Gamma 2'ffi_ ; therefore f 2 \Gamma (U; O(KX) \Omega J (')) if and only if _ \Lambda f 2 \Gamma (_ \Gamma 1 (U ); O(KX ) \Omega J (' ffi _)). This proves Prop. 15.5. \Lambda 15.6. Remark. If ' has algebraic or analytic singularities (cf. definition 11.7), the calculation of J (') is reduced to a purely algebraic problem. The first observation is that J (') is easily calculated if ' = P ffj log jgjj where Dj = g \Gamma 1 j (0) are smooth irreducible divisors with normal crossings. Then J (') is the sheaf of holomorphic functions h on the open set U ae X, satisfying Z U jhj 2 Y jgjj \Gamma 2ffj dV ! +1: Since the gj can be taken as coordinate functions in suitable local coordinate systems (z1; : : : ; zn), the integrability condition is that h is divisible by Q g mj j , where mj \Gamma ffj ? \Gamma 1 for each j, i.e. mj * bffjc (where b c denotes the integral part). Consequently J (') = O(\Gamma bDc) = O(\Gamma X bffjcDj) where bDc is the integral part of the Q-divisor D = P ffjDj. Now consider the general case of algebraic or analytic singularities and assume that ' , ff 2 log \Gamma jf1j 2 + \Delta \Delta \Delta + jfN j 2 \Delta in a neighbourhood of the poles. According to the remark stated after definition 11.7, we can assume that the (fj) are generators of the sheaf of integrally closed ideals J = J('=ff), defined as the sheaf of holomorphic functions h such that jhj ^ C exp('=ff). In this case, the calculation is done as follows. Let us first choose a smooth modification _ : ~ X ! X of X such that _ \Lambda J is an invertible sheaf O(\Gamma D) associated to a divisor with normal crossings D = P *jDj, where (Dj) are the components of the exceptional divisor of ~ X. (Consider the blow-up X 0 of X along the ideal J, so that the inverse image of J on X 0 becomes an invertible sheaf O(\Gamma D 0 ), then blow-up X 0 again so as to render X 0 smooth and D 0 with normal crossings, by invoking Hironaka [Hi64].) We then have K ~ X = 75 15. vanishing theorems of nadel and kawamata-viehweg _ \Lambda KX + R where R = P aejDj is the divisor of zeros of the jacobian J_ of the blow-up map. From the direct image formula 15.5, we deduce J (') = _\Lambda \Gamma O(K ~ X \Gamma _ \Lambda KX ) \Omega J (' ffi _) \Delta = _\Lambda \Gamma O(R) \Omega J (' ffi _) \Delta : Now the (fj ffi _) are generators of the ideal O(\Gamma D), therefore ' ffi _ , ff X *j log jgjj where the gj are local generators of O(\Gamma Dj). We are thus reduced to calculating the multiplier ideal sheaf in the case where the poles are given by a Q-divisor with normal crossings P ff*jDj. We obtain J (' ffi _) = O(\Gamma P bff*jcDj), therefore J (') = _\Lambda O ~ X ( X (aej \Gamma bff*jc)Dj): \Lambda 15.7. Exercise. Calculate the multiplier ideal sheaf J (') associated to the psh function ' = log(jz1j ff1 + \Delta \Delta \Delta + jzpj ffp ), for arbitrary real numbers ffj ? 0. Indication. By using Parseval's formula and polar coordinates zj = rj e i\Theta j , show that the problem is equivalent to determining for which p-tuples (fi1; : : : ; fip) 2 N p the integral Z [0;1]p r 2fi1 1 \Delta \Delta \Delta r 2fip p r1dr1 \Delta \Delta \Delta rpdrp r 2ff1 1 + \Delta \Delta \Delta + r 2ffp p = Z [0;1]p t (fi1+1)=ff1 1 \Delta \Delta \Delta t (fip+1)=ffp p t1 + \Delta \Delta \Delta + tp dt1 t1 \Delta \Delta \Delta dtp tp is convergent. Deduce from this that J (') is generated by the monomials z fi1 1 \Delta \Delta \Delta z fip p such that P (fip + 1)=ffp ? 1. (This exercise shows that the analytic definition of J (') is also sometimes very convenient for calculations). \Lambda Let E be a line bundle over X with a given singular metric h with curvature current \Theta h(E). If ' is the weight representing the metric h on an open set \Omega ae X, the sheaf of ideals J (') is independent of the choice of the trivialization. It is therefore the restriction to \Omega of a global coherent sheaf on X that we will denote by J (h) = J ('), by abuse of notation. In this context, we have the following fundamental vanishing theorem, which is probably one of the most central results in algebraic or analytic geometry. (As we will see later, this theorem contains the Kawamata-Viehweg vanishing theorem as a special case.) 15.8. Nadel Vanishing Theorem ([Nad89], [Dem93b]). Let (X; !) be a weakly pseudoconvex K"ahler manifold, and let E be a holomorphic line bundle on X with a given singular Hermitian metric h of weight '. Assume that there exists a positive continuous function ffl on X such that i \Theta h(E) * ffl!. Then H q \Gamma X; O(KX + E) \Omega J (h) \Delta = 0 for all q * 1: Proof. Let L q be the sheaf of germs of (n; q)-forms u with values in E and with measurable coefficients, for which juj 2 e \Gamma 2' and jd 00 uj 2 e \Gamma 2' are simultaneously locally integrable. The operator d 00 defines a complex of sheaves (L ffl ; d 00 ) which is a resolution of the sheaf O(KX + E) \Omega J ('): Indeed, the kernel of d 00 in degree 0 consists of the germs of holomorphic n-forms with values in E which satisfy the integrability condition. Therefore the coefficient function belongs to J ('), and the exactness at degree q * 1 arises from Corollary 14.3 applied to arbitrary small balls. Since each sheaf L q is a C 1 -module, L ffl is a resolution by acyclic sheaves. Let j.-p. demailly, part ii: L 2 estimations and vanishing theorems 76 be a C 1 psh exhaustion function on X. We apply Corollary 14.3 globally on X, with the initial metric of E multiplied by the factor e \Gamma O/ffi , where O/ is an increasing convex function of arbitrary growth at infinity. This factor can be used to ensure convergence of integrals at infinity. From Corollary 14.3, we then deduce that H q (\Gamma (X; L ffl )) = 0 for q * 1. The theorem follows by virtue of the de Rham-Weil Isomorphism Theorem (1.2). \Lambda 15.9. Corollary. Let (X; !); E and ' be given as in Theorem 15.8, and assume given x1; : : : ; xN isolated points of the variety of zeros V (J (')). Then there exists a surjective map H 0 (X; O(KX + E)) \Gamma ! M 1^j^N O(KX + E)x j \Omega \Gamma OX =J (') \Delta xj : Proof. Consider the long exact cohomology sequence associated to the short exact sequence 0 ! J (') ! OX ! OX =J (') ! 0, twisted by O(KX + E), and apply Theorem 15.8 to obtain the vanishing of the first group H 1 . The stated surjective property follows. \Lambda 15.10. Corollary. Let (X; !); E and ' be given as in Theorem 15.8. Assume that the weight function ' satisfies *('; x) * n + s at a given point x 2 X for which *('; y) ! 1, for y 6= x close enough to x. Then H 0 (X; KX + E) generates all the s-jets of sections at the point x. Proof. Skoda's Lemma 15.3 b) shows that e \Gamma 2' is integrable in a neighbourhood of any point y 6= x sufficiently close to x, therefore J (')y = OX;y, whereas J (')x ae m s+1 X;x according to 15.3 a). Corollary 15.10 is therefore a special case of 15.9. \Lambda The philosophy of the results (which can be regarded as generalization of the H"ormander-Bombieri-Skoda Theorem [Bom70], [Sko72,75]), is that the problem of constructing holomorphic sections of KX + E can be solved by constructing suitable Hermitian metrics on E such that the weight ' has isolated logarithmic points at the given points xj . 15.11. Exercise. Assume that X is compact and that L is a positive line bundle on X. Let fx1; : : : ; xN g be a finite set. Show that there exists constants a; b * 0 depending only on L and N such that for any s 2 N, the group H 0 (X; O(mL)) generates the jets of order s at any point xj, for m * as + b. Indication. Apply Corollary 15.9 to E = \Gamma KX + mL, with a singular metric on L of the form h = h0e \Gamma ffl , where h0 is C 1 with positive curvature, ffl ? 0 small, and (z) , log jz \Gamma xj j in a neighbourhood of xj . Deduce from this the Kodaira embedding theorem: 15.12. Kodaira Embedding Theorem. If L is a line bundle on a compact complex manifold, then L is ample if and only if L is positive. \Lambda An equivalent way to state the Kodaira embedding theorem is the following: 15.13. Kodaira criterion for projectivity. A compact complex manifold X is projective algebraic if and only if X contains a Hodge metric. That is, a K"ahler metric with integral cohomology class. 77 15. vanishing theorems of nadel and kawamata-viehweg Proof. If X ae P N is projective algebraic, then the restriction of the FubiniStudy metric to X is a Hodge metric. Conversely, if X has a Hodge metric !, the cohomology class representative f!g in H 2 (X; Z) defines a complex topological (i.e. C 1 ) line bundle, say L. Since ! is of type (1; 1), the exponential exact sequence (8.20) H 1 (X; O \Lambda X ) ! H 2 (X; Z) ! H 2 (X; O) = H 0;2 (X; C ) shows that the line bundle L can be represented by a cocycle in H 1 (X; O \Lambda X ). In other words, L is endowed with a complex structure. Moreover, there exists a Hermitian metric h on L such that i 2ss \Theta h(L) = !. Consequently, L is ample and X is projective algebraic. 15.14. Exercise (Riemann conditions characterizing Abelian varieties). A complex torus X = C n =\Gamma is called an Abelian variety if X is projective algebraic. Show by using (15.13) that a torus X is an Abelian variety if and only if there exists a positive definite Hermitian form H on C n such that Im H(fl1; fl2) 2 Z for all fl1; fl2 in the lattice \Gamma . Indication. Use a process of averaging to reduce the proof to the case of K"ahler metric invariant by translations. Observe that the real torus Zfl1 + Zfl2 defines a system of generators of the homology group H2(X; Z) and that R Zfl1+Zfl2 ! = !(fl1; fl2). 15.15. Exercise (solution of the Levi problem). Show that the following two properties are equivalent. a) X is strongly pseudoconvex, i.e. X admits a strongly psh exhaustion function. b) X is a Stein, i.e. the global holomorphic functions separate points, furnishing a system of local coordinates at every point, and X is holomorphically convex. (By definition, this means that for any discrete sequence (z*) in X, there exists a function f 2 H 0 (X; OX) such that jf (z*)j ! 1.) \Lambda 15.16. Remark. As long as one is interested only in the case of forms of bidegree (n; q); n = dim X, the L 2 estimates extend to the complex spaces acquiring arbitrary singularities. Indeed, if X is a complex space and ' a psh weight function on X, one can still define a sheaf KX(') on X, such that the sections of KX (') on an open set U are the holomorphic n-forms f on the regular part U " Xreg, satisfying the integrability condition i n 2 f ^ fe \Gamma 2' 2 L 1 loc (U ). In this context, the functorial property 15.5 can be written (or is written) _\Lambda (KX0(' ffi _)) = KX ('); and it is valid for arbitrary complex spaces X; X 0 , _ : X 0 ! X being a modification. If X is non-singular, one has KX (') = O(KX) \Omega J ('), however, if X is singular, the symbols KX and J (') do not have to be dissociated. The statement of the Nadel vanishing theorem becomes H q (X; O(E) \Omega KX(')) = 0 for q * 1, under the same hypothesis (X K"ahler and weakly pseudoconvex, curvature of E * ffl!). The proof is obtained by restricting all the situations to Xreg. Although in general Xreg is not weakly pseudoconvex (a necessary condition being codimXsing = 1), Xreg is always K"ahlerian complete (the complement of an analytic subset in a weakly pseudoconvex K"ahler space is K"ahlerian complete, see for example [Dem82]). As a consequence, the Nadel vanishing theorem is essentially insensitive to the presence of singularities. \Lambda j.-p. demailly, part ii: L 2 estimations and vanishing theorems 78 We now deduce an algebraic version of the Nadel vanishing theorem obtained independently by Kawamata [Kaw82] and Viehweg [Vie82]. (The original proof relies on a different method using cyclic coverings to reduce to the case situation of the ordinary Kodaira Theorem.) Before stating the theorem, we need a definition. 15.17. Definition. A line bundle L on a compact complex manifold is called large if its Kodaira dimension is equal to n = dim X, that is, if there exists a constant c ? 0 such that dim H 0 (X; O(kL)) * ck n ; k * k0: 15.18. Definition. A line bundle L on a projective algebraic manifold is called numerically effective (nef for short) if L satisfies one of the following three equivalent properties: a) For any irreducible algebraic curve C ae X, one has L \Delta C = R C c1(L) * 0. b) If A is an ample line bundle, then kL + A is ample for all k * 0. c) For any ffl ? 0, there exists a C 1 Hermitian metric hffl on L such that \Theta h ffl (L) * \Gamma ffl!, where ! is a fixed Hermitian metric on X. The equivalence of properties 15.18 a) and b) is well-known and we will omit it here (see for example Hartshorne [Har70] for the proof). It is clear in addition that 15.18 c) implies 15.18 a), while 15.18 b) implies 15.18 c). Indeed if ! = i 2ss \Theta (A) is the curvature of a metric of A with positive curvature, and if hk is a metric on L inducing a metric with positive curvature on kL + A, it becomes k i 2ss \Theta (L) + i 2ss \Theta (A) ? 0, where i 2ss \Theta (L) * \Gamma 1 k !. Now, if D = P ffjDj * 0 is an effective Qdivisor, we define the multiplier ideal sheaf J (D) to be the sheaf J (') associated to the psh function ' = P ffj log jgjj defined by the generators gj of O(\Gamma Dj). According to remark 15.6, the calculation of J (D) can be done algebraically by making use of desingularizations _ : ~ X ! X such that _ \Lambda D becomes a divisor with normal crossings on ~ X. 15.19. Kawamata-Viehweg Vanishing Theorem. Let X be a projective algebraic manifold, and let F be a line bundle on X such that a multiple mF of F can be written in the form mF = L + D, where L is a nef and large line bundle, and D an effective divisor. Then H q \Gamma X; O(KX + F ) \Omega J (m \Gamma 1 D) \Delta = 0 for q * 1: 15.20. Corollary. If F is nef and large, then H q (X; O(KX + F )) = 0 for q * 1. Proof. Let A be a non-singular very ample divisor. There is an exact sequence 0 ! H 0 (X; O(kL \Gamma A)) ! H 0 (X; O(kL)) ! H 0 (A; O(kL)_A); and dim H 0 (A; O(kL)_A) ^ Ck n\Gamma 1 for a certain constant C * 0. Since L is large, there exists an integer k0 AE 0 such that O(k0L \Gamma A) has a non-trivial section. If E is the divisor of this section, we have O(k0L \Gamma A) ' O(E), therefore O(k0L) ' O(A + E). Now, for k * k0, we arrive at O(kL) = O((k \Gamma k0)L + A + E). According to 15.18 b), the line bundle O((k \Gamma k0)L + A) is ample, therefore it comes with a C 1 Hermitian metric hk = e \Gamma 'k , and with positive definite curvature form !k = i 2ss \Theta ((k \Gamma k0)L + A). Let 'D = P ffj log jgjj be the weight of the singular metric on O(D) described in example 11.21, such that i 2ss \Theta (O(D)) = [D], and in a 79 16. on the conjecture of fujita similar way, let 'E be the weight such that i 2ss \Theta (O(E)) = [E]. We define a singular metric on O(kL) = O((k \Gamma k0)L + A + E) by means of the weight 'k + 'E, and then we obtain a singular metric on O(mF ) = O(L + D), by considering the weight 1 k ('k + 'E) + 'D. Finally, we obtain a metric on F of weight 'F = 1 km ('k + 'E) + 1 m 'D: The corresponding curvature form is i 2ss \Theta (F ) = 1 km (!k + [E]) + 1 m [D] * 1 km !k ? 0: Moreover 'F has algebraic singularities, and by taking k sufficiently large we have J ('F ) = J ` 1 km E + 1 m D ' = J ` 1 m D ' : Indeed, J ('F ) is calculated by taking the integral part of a Q-divisor with normal crossings, obtained by the means of a suitable modification (as was explained in remark 15.6). The divisor 1 km E + 1 m D furnishes therefore the same integral part as 1 m D when k is large. The Nadel Theorem then implies the desired vanishing result for all q * 1. \Lambda 16. On the conjecture of Fujita Given an ample line bundle L, a fundamental question is of determining an effective integer m0 such that mL is very ample for m * m0. The example where X is a hyperelliptic curve of genus g and where L = O(p) is associated to one of the 2g+2 Weierstrass points, shows that m0 must be at least equal to 2g+1 (additionally it is checked rather easily that m0 = 2g +1 always answers the question for a curve). It follows from this that m0 must necessarily depend on the geometry of X, and cannot depend only on the dimension of X. However, when mL is replaced by the "adjoint" line bundle KX + mL, a simple universal answer seems likely to emerge. 16.1. Fujita's conjecture ([Fuj87]). If L is an ample line bundle on a projective manifold of dimension n, then i) KX + (n + 1)L is generated by its global sections; ii) KX + (n + 2)L is very ample. The bounds predicted by the conjecture are optimal for (X; L) = (P n ; O(1)), since in this case KX = O(\Gamma n \Gamma 1). The conjecture is easy to verify in the case of curves (exercise!), and I. Reider [Rei88] has solved the conjecture in the affirmative in the case n = 2. Ein-Lazarsfeld [EL93] and Fujita [Fuj93] arrived at establishing part i) in dimension 3, and a very thorough refinement of their technique allowed Kawamata [Kaw95] to also arrive at the case of dimension 4 1 . The other cases of the conjecture, namely i) for n * 5 and ii) for n * 3, remain for the time being unsolved. The first step in the direction of this conjecture for arbitrary dimension n has been realized in 1991 (work published 2 years later in [Dem93]), by means of an analytic method relying on a resolution of a Monge-Ampere equation. Similar results were obtained by Koll'ar [Kol92] employing entirely algebraic 1 The technique of Fujita [Fuj93] and Kawamata [Kaw95] has just been simplified considerably and clarified by S. Helmke [Hel96]. j.-p. demailly, part ii: L 2 estimations and vanishing theorems 80 methods. We refer to [Laz93] for an excellent article devoted to the synthesis of these developments, as well as [Dem94] for the analytic version of the theory. This section is devoted to the proof of some results dependent on Kujita's conjecture in arbitrary dimension. The principal ideas of interest here are inspired by some recent work of Y.T. Siu [Siu96]. Siu's method, which is naturally algebraic and relatively elementary, consists of combining the Riemann-Roch formula with the Kawamata-Viehmeg vanishing theorem (however, it will be much more convenient to use this Nadel's formulation of the theorem, using the multiplier ideal sheaves). Subsequently, X will denote a projective algebraic manifold of dimension n. The first useful observation is the following classical consequence of the Riemann-Roch formula: 16.2. Particular case of the Riemann-Roch formula. Let J ae OX be a coherent sheaf of ideals on X such that the variety of zeros V (J) is of dimension d (with possibly some components of lower dimension). Let Y = P *jYj be the effective algebraic cycle of dimension d associated to the components of dimension d of V (J) (the multiplicities *j taking into account the multiplicity of the length of the ideal J along each component). Then, for any line bundle E, the Euler characteristic O/(X; O(E + mL) \Omega OX =O(J)) is a polynomial P (m) of degree d and with leading coefficient L d \Delta Y=d! \Lambda The second useful fact is an elementary lemma concerning the numerical polynomials (polynomials with rational coefficients, defining a map of Z into Z). 16.3. Lemma. Let P (m) be a numerical polynomial of degree d ? 0 and with leading coefficient ad=d!; ad 2 Z; ad ? 0. We assume that P (m) * 0 for all m * m0. Then a) For all N * 0, there exists m 2 [m0; m0 + N d] such that P (m) * N . b) For all k 2 N, there exists m 2 [m0; m0 + kd] such that P (m) * adk d =2 d\Gamma 1 . c) For all N * 2d 2 , there exists m 2 [m0; m0 + N ] such that P (m) * N . Proof. a) Each one of the N equations P (m) = 0; P (m) = 1; : : : ; P (m) = N \Gamma 1 has at most d roots, therefore there is necessarily an integer m 2 [m0; m0+dN ] which is not a root of these equations. b) By virtue of Newton's formula for the iterated differences \Delta P (m) = P (m + 1) \Gamma P (m), we obtain \Delta d P (m) = X 1^j^d (\Gamma 1) j ` d j ' P (m + d \Gamma j) = ad; 8m 2 Z: Consequently, if j 2 \Phi 0; 2; 4; : : : ; 2b d=2c \Psi ae [0; d] is the even integer realizing the maximum of P (m0 + d \Gamma j) on this finite set, we obtain 2 d\Gamma 1 P (m0 + d \Gamma j) = `` d 0 ' + ` d 2 ' + \Delta \Delta \Delta ' P (m0 + d \Gamma j) * ad; whereby we obtain the existence of an integer m 2 [m0; m0 + d] with P (m) * ad=2 d\Gamma 1 . The result is therefore proven for k = 1. In the general case, we apply this particular result to the polynomial Q(m) = P (km \Gamma (k \Gamma 1)m0), for which the leading coefficient is adk d =d! 81 16. on the conjecture of fujita c) If d = 1, part a) already gives the result. If d = 2, a glance at the parabola shows that max m2[m0;m0+N] P (m) * ae a2N 2 =8 if N is even, a2(N 2 \Gamma 1)=8 if N is odd; therefore maxm2[m 0;m0+N] P (m) * N whenever N * 8. If d * 3, we apply b) with k equal to the smallest integer satisfying k d =2 d\Gamma 1 * N , i.e. k = d2(N=2) 1=d e, where dxe 2 Z denotes the greater integer. Then kd ^ (2(N=2) 1=d + 1)d ^ N so long as N * 2d 2 , as one sees after a short calculation. \Lambda We now apply the Nadel vanishing theorem in an analogous way to that of Siu [Siu96], with some simplifications in the technique and some improvements for the bounds. Their method simultaneously gives a simple proof of a fundamental classical result due to Fujita. 16.4. Theorem (Fujita). If L is an ample line bundle on a projective manifold X of dimension n, then KX + (n + 1)L is nef. Using the theory of Mori and the "base point free theorem" ([Mor82], [Kaw84]), one can show in fact that KX + (n + 1)L is semi-ample, and that there exists a positive integer m such that m(KX + (n + 1)L) is generated by its sections (see [Kaw85] and [Fuj87]). The proof is based on the observation that n + 1 is the maximum length of the extremal rays of smooth projective varieties of dimension n. Their proof of (16.4) is different and was obtained at the same time as the proof of th. (16.5) below. 16.5. Theorem. Let L be an ample line bundle and let G be a nef line bundle over a projective manifold X of dimension n. Then the following properties hold. a) 2KX + mL + G simultaneously generates the jets of order s1; : : : ; sp 2 N at arbitrary points x1; : : : ; xp 2 X, i.e., there exists a surjective map H 0 (X; O(2KX + mL + G)) i M 1^j^p O(2KX + mL + G) \Omega OX;x j =m sj+1 X;xj ; so long as m * 2 + P 1^j^p \Gamma 3n+2sj\Gamma 1 n \Delta . In particular 2KX + mL + G is very ample for m * 2 + \Gamma 3n+1 n \Delta . b) 2KX + (n + 1)L + G simultaneously generates the jets of order s1; : : : ; sp at arbitrary points x1; : : : ; xp 2 X so long as the intersection numbers L d \Delta Y of L on all the algebraic subsets Y of X of dimension d are such that L d \Delta Y ? 2 d\Gamma 1 bn=dc d X 1^j^p ` 3n + 2sj \Gamma 1 n ' : Proof. The proofs of (16.4) and (16.5a, b) are completely parallel, that is why we will present them simultaneously (in the case of (16.4), it is simply agreed that fx1; : : : ; xpg = ;). The idea is to find an integer (or a rational number) m0 and a singular Hermitian metric h0 on KX + m0L for which the curvature current is strictly positive, \Theta h 0 * ffl!, such that V (J (h0)) is of dimension 0 and such that the weight '0 of h0 satisfies *('0; xj ) * n + sj for all j. Since L and G are j.-p. demailly, part ii: L 2 estimations and vanishing theorems 82 nefs, 15.18 c) implies that (m \Gamma m0)L + G has for all m * m0 a metric h 0 for which the curvature \Theta h0 has an arbitrarily small negative part, say \Theta h0 * \Gamma ffl 2 !. Then \Theta h 0 + \Theta h0 * ffl 2 ! is positive definite. An application of Cor. 15.9 to F = KX + mL + G = (KX + m0L) + ((m \Gamma m0)L + G) with metric h0 \Omega h 0 guarantees the existence of sections of KX + F = 2KX + mL + G producing the desired jets for m * m0. Fix an embedding \Phi j_Lj : X ! P N ; _ AE 0, given by the sections *0; : : : ; *N 2 H 0 (X; _L), and let hL be the associated metric on L, with positive definite curvature form ! = \Theta (L). To obtain the desired metric h0 on KX + m0L, one fixes an integer a 2 N \Lambda and one uses a process of double induction to construct singular metrics (hk;*)**1 on aKX + bkL, for a decreasing sequence of positive integers b1 * b2 * \Delta \Delta \Delta * bk * \Delta \Delta \Delta . Such a sequence is necessarily stationary and m0 will be precisely the stationary limit m0 = lim bk=a. The metrics hk;* are chosen to be the type that satisfy the following properties: ff) hk;* is an "algebraic" metric of the form jj,jj 2 hk;* = jo/k(,)j 2 ( P 1^i^*;0^j^N jo/ (a+1)_ k (oe a_ i \Delta * (a+1)bk\Gamma ami j )j 2 ) 1=(a+1)_ ; defined by the sections oei 2 H 0 \Gamma X; O((a+1)KX +miL) \Delta ; mi ! a+1 a bk; 1 ^ i ^ *, where , 7! o/k(,) is an arbitrary local trivialization of aKX + bkL. Observe that oe a_ i \Delta * (a+1)bk\Gamma ami j is a section of a_((a + 1)KX + miL) + ((a + 1)bk \Gamma ami)_L = (a + 1)_(aKX + bkL): fi) ordx j (oei) * (a + 1)(n + sj) for all i; j; fl) J (hk;*+1) oe J (hk;* ) and J (hk;*+1) 6= J (hk;*) as long as the variety of zeros V (J (hk;* )) is positive dimensional. The weight 'k;* = 1 2(a+1)_ log P fi fi o/ (a+1)_ k (oe a_ i \Delta * (a+1)bk\Gamma ami j ) fi fi 2 of hk;* is plurisubharmonic and the condition mi ! a+1 a bk implies (a + 1)bk \Gamma ami * 1, therefore the difference 'k;* \Gamma 1 2(a+1)_ log P jo/ (*j )j 2 is also plurisubharmonic. Consequently i 2ss \Theta h k;* (aKX + bkL) = i ss d 0 d 00 'k;* * 1 (a+1) !. Moreover, condition fi) clearly implies that *('k;* ; xj) * a(n + sj). Finally, condition fl) combined with the strong Noetherian property of coherent sheaves guarantees that the sequence (hk;*)**1 will eventually produce a subscheme V (J (hk;*)) of dimension 0. One can check that the sequence (hk;* )**1 terminates at this point, and we set hk = hk;* to be the final metric thus reached, such that dim V (J (hk)) = 0. For k = 1, it is clear that the desired metrics (h1;*)**1 exist if b1 is chosen large enough. (For example, such that (a + 1)KX + (b1 \Gamma 1)L generates the jets of order (a + 1)(n + max sj) at every point. Then the sections oe1; : : : ; oe* can be chosen such that m1 = \Delta \Delta \Delta = m* = b1 \Gamma 1.) We assume that the metrics (hk;*)**1 and hk are already constructed, and proceed with the construction of (hk+1;*)**1. We use again induction on *, and assume that hk+1;* is already constructed and that dim V (J (hk+1;* )) ? 0. We begin our induction with * = 0, and let us declare in this case that J (hk+1;0) = 0 (this corresponds to an infinite metric of weight identically equal to \Gamma 1). By virtue of the Nadel vanishing theorem applied to Fm = aKX + mL = (aKX + bkL) + (m \Gamma bk)L for the metric hk \Omega (hL) \Omega m\Gamma bk , we 83 16. on the conjecture of fujita obtain H q (X; O((a + 1)KX + mL) \Omega J (hk)) = 0 for q * 1; m * bk: Since V (J (hk)) is of dimension 0, the sheaf OX=J (hk) is a skyscraper sheaf and the exact sequence 0 ! J (hk) ! OX ! OX=J (hk) ! 0 twisted by the invertible sheaf O((a + 1)KX + mL) shows that H q (X; O((a + 1)KX + mL)) = 0 for q * 1; m * bk: Analogously, we find H q (X; O((a + 1)KX + mL) \Omega J (hk+1;* )) = 0 for q * 1; m * bk+1 (it is therefore true for * = 0, since J (hk+1;0) = 0), and when m * max(bk; bk+1) = bk; the exact sequence 0 ! J (hk+1;*) ! OX ! OX =J (hk+1; *) ! 0 implies H q (X; O((a + 1)KX + mL) \Omega OX =J (hk+1;*)) = 0 for q * 1; m * bk: In particular, since the group H 1 above is zero, any section u 0 of (a + 1)KX + mL on the sub-scheme V (J (hk+1;*)) has an extension u to X. Fix a basis u 0 1 ; : : : ; u 0 N of sections of this sheaf on V (J (hk+1;* )) and take arbitrary extensions u1; : : : ; uN to X. Consider the linear map allotting to each section u on X the collection of jets of order (a + 1)(n + sj) \Gamma 1 at the points xj, i.e. u = X 1^j^N ajuj 7! M J (a+1)(n+sj)\Gamma 1 xj (u): Since the rank of the bundle of s-jets is \Gamma n+s n \Delta , the target space is of dimension ffi = X 1^j^p ` n + (a + 1)(n + sj) \Gamma 1 n ' : To obtain a section oe*+1 = u satisfying condition fi) and having a non-trivial restriction oe 0 *+1 to V (J (hk+1;* )), we need at least N = ffi + 1 independent sections u 0 1 ; : : : ; u 0 N . This condition will be realized by applying Lemma (16.3) to the numerical polynomial P (m) = O/(X; O((a + 1)KX + mL) \Omega OX =J (hk+1;* )) = h 0 (X; O((a + 1)KX + mL) \Omega OX=J (hk+1;*)) * 0; m * bk: The polynomial P is of degree d = dim V (J (hk+1;*)) ? 0. We therefore obtain the existence of an integer m 2 [bk; bk + j] such that N = P (m) * ffi + 1, for some explicit integer j 2 N. (For example, j = n(ffi + 1) is always appropriate according to (16.3 a), but it will be equally important to use the other possibilities to optimize the choices.) We then find a section oe*+1 2 H 0 (X; (a + 1)KX + mL) having a nontrivial restriction oe 0 *+1 to V (J (hk+1;*)), vanishing to order * (a + 1)(n + sj) at each point xj . Now set m*+1 = m, and the condition m*+1 ! a+1 a bk+1 is realized if bk + j ! a+1 a bk+1. This shows that one can choose recursively bk+1 = _ a a + 1 (bk + j) * + 1: j.-p. demailly, part ii: L 2 estimations and vanishing theorems 84 By definition, hk+1;* ^ hk+1;* , therefore J (hk+1;*+1) oe J (hk+1;* ). It is the case that J (hk+1;*+1) 6= J (hk+1;*), because J (hk+1;*+1) contains the sheaf of ideals associated to the divisor of zeros of oe*+1, whereas oe*+1 is not identically zero on V (J (hk+1;*)). Now, an easy calculation shows that the iterated sequence bk+1 = b a a+1 (bk + j)c + 1 stabilizes to the limit value bk = a(j + 1) + 1, for any initial value b1 greater than this limit. In this way, we obtain a metric h1 with positive definite curvature on aKX + (a(j + 1) + 1)L, such that dim V (J (h1)) = 0 and *('1; xj) * a(n + sj) at each point xj . Proof of (16.4). In this case, the set fxjg is taken to be the empty set, therefore ffi = 0. By virtue of (16.3 a), the condition P (m) * 1 is realized for at least one integer m 2 [bk; bk + n], therefore one can take j = n. Since _L is very ample, _L has a metric having an isolated logarithmic pole of Lelong number 1 at each given point (for example, the algebraic metric defined by the sections of _L vanishing at x0). Therefore F 0 a = aKX + (a(n + 1) + 1)L + n_L has a metric h 0 a such that V (J (h 0 a )) is of dimension zero and contains fx0g. By virtue of Cor. (15.9), we conclude that KX + F 0 a = (a + 1)KX + (a(n + 1) + 1 + n_)L is generated by its sections, in particular KX + a(n+1)+1+n_ a+1 L is nef. By letting a tend to +1, we deduce that KX + (n + 1)L is nef. \Lambda Proof of (16.5 a). It suffices here to choose a = 1. Then ffi = X 1^j^p ` 3n + 2sj \Gamma 1 n ' : If fxjg 6= ;, one has ffi + 1 * \Gamma 3n\Gamma 1 n \Delta + 1 * 2n 2 for n * 2. Lemma (16.3 c) shows that P (m) * ffi + 1 for at least one m 2 [bk; bk + j] with j = ffi + 1. We begin the induction procedure k 7! k + 1 with b1 = j + 1 = ffi + 2, because the only necessary property for the induction step is the vanishing property H q (X; 2KX + mL) = 0 for q * 1; m * b1; which is realized according to Kodaira's vanishing theorem and the ampleness property of KX + b1L. (We use here the result of Fujita (16.4), by observing that b1 ? n+1.) The recursive formula bk+1 = b 1 2 (bk+j)c+1 then gives bk = j+1 = ffi+2 for all k, and (16.5 a) follows. \Lambda Proof of (16.5 b). Completely similar to (16.5 a), except that we choose j = n; a = 1 and bk = n + 1 for all k. By applying Lemma (16.3 b), we have P (m) * adk d =2 d\Gamma 1 for at least one integer m 2 [m0; m0 + kd], where ad ? 0 is the leading degree coefficient of P . By virtue of Lemma (16.2), we have ad * infdim Y =d L d \Delta Y . Take k = bn=dc. The condition P (m) * ffi + 1 can then be realized for an integer m 2 [m0; m0 + kd] ae [m0; m0 + n], provided that inf dim Y =d L d \Delta Y bn=dc d =2 d\Gamma 1 ? ffi; that which is equivalent to the condition in (16.5 b). \Lambda 85 17. an effective version of matsusaka's big theorem The big disadvantage of the described technique is that one must necessarily utilize multiples of L to avoid the zeros of the Hilbert polynomial, in particular it is not possible to directly obtain a criterion of large ampleness for 2KX + L in the statement of (16.5 b). Such a criterion can nevertheless be obtained with the aid of the following elementary lemma. 16.6. Lemma. Suppose that there exists an integer _ 2 N \Lambda such that _F simultaneously generates all the jets of order _(n + sj) + 1 at every point xj of a subset fx1; : : : ; xpg ae X. Then KX + F simultaneously generates all the jets of order sj at the point xj. Proof. Choose the algebraic metric on F defined by a basis oe1; : : : ; oeN of the space of sections of _F which vanish to order _(n + sj) + 1 at each point xj . Since we are still free to choose the homogenous term of degree _(n + sj) + 1 in the Taylor expansion of these sections at the points xj , we see that x1; : : : ; xp are isolated zeros of "oe \Gamma 1 j (0). If ' is the weight of the metric of F about xj, we therefore have '(z) , (n + sj + 1 _ ) log jz \Gamma xjj in suitable coordinates. Replace ' in a neighbourhood of xj by ' 0 (z) = max \Gamma '(z); jzj 2 \Gamma C + (n + sj) log jz \Gamma xj j \Delta and we leave ' unchanged everywhere else (this is possible by taking C ? 0 sufficiently large). Then ' 0 (z) = jzj 2 \Gamma C + (n + sj) log jz \Gamma xjj in a neighbourhood of xj, in particular ' 0 is strictly plurisubharmonic near xj. In this way, we obtain a metric h 0 on F with semi-positive curvature everywhere on X, and has positive definite curvature in a neighbourhood of fx1; : : : ; xpg. The resulting conclusion then is a direct application of the L 2 estimates (14.2). \Lambda 16.7. Theorem. Let X be a projective manifold of dimension n and L an ample line bundle on X. Then 2KX + L simultaneously generates the jets of order s1; : : : ; sp at arbitrary points x1; : : : ; xp 2 X so long as the intersection numbers L d \Delta Y of L on all the algebraic subsets Y ae X of dimension d satisfy L d \Delta Y ? 2 d\Gamma 1 bn=dc d X 1^j^p ` (n + 1)(4n + 2sj + 1) \Gamma 2 n ' ; 1 ^ d ^ n: Proof. Lemma (16.6) applied with F = KX + L and _ = n + 1 shows that the desired property for the jets of 2KX + L occurs if (n + 1)(KX + L) generates the jets of order (n + 1)(n + sj) + 1 at the points xj . Lemma (16.6) applied again with F = pKX + (n + 1)L and _ = 1 shows by descending induction on p that it suffices that F generates all the jets of order (n + 1)(n + sj) + 1 + (n + 1 \Gamma p)(n + 1) at the points xj. In particular, for 2KX + (n + 1)L it suffices to obtain all the jets of order (n + 1)(2n + sj \Gamma 1) + 1. Th. (16.5 b) then gives the desired condition. \Lambda We conclude by mentioning some immediate consequences of th. 16.5, obtained by taking L = \Sigma KX. 16.8. Corollary. Let X be a projective manifold of general type, with KX ample and dim X = n. Then mKX is very ample for m * m0 = \Gamma 3n+1 n \Delta + 4. 16.9. Corollary. Let X be a Fano variety (that is, a projective manifold such that \Gamma KX is ample), of dimension n. Then \Gamma mKX is very ample for m * m0 = \Gamma 3n+1 n \Delta . j.-p. demailly, part ii: L 2 estimations and vanishing theorems 86 17. An effective version of Matsusaka's big theorem We encounter here the problem of finding an explicit integer m0 such that mL is very ample for m * m0. The existence of such a bound m0, depending only on the dimension and the coefficients of the Hilbert polynomial of L, was first established by Matsusaka [Mat72]. Further Koll'ar and Matsusaka [KoM83] have shown that one could indeed find a bound m0 = m0(n; L n ; KX \Delta L n\Gamma 1 ) dependent only on n = dim X and on the first two coefficients. Recently, Siu [Siu93] has obtained an effective version of the same result furnishing an explicit "reasonable" bound m0 (although this bound is unfortunately still far from being optimal). We explain here the method of Siu, starting from some simplifications and improvements suggested in [Dem96]. The starting point is the following lemma. 17.1. Lemma. Let F and G be nef line bundles on X. If F n ? nF n\Gamma 1 \Delta G, then any positive multiple k(F \Gamma G) admits a non-trivial section for k * k0 sufficiently large. Proof. The lemma can be proven as a special case of the holomorphic Morse inequalities (see [Dem85], [Tra91], [Siu93], [Ang95]). We give here a simple proof, following a suggestion of F. Catanese. We can assume that F and G are very ample (if not, it suffices to replace F and G by F 0 = pF + A and G 0 = pG + A with A very ample and sufficiently positive to ensure large ampleness of any sum with an nef bundle, then to choose p ? 0 large enough for which F 0 and G 0 satisfy the same numerical hypothesis as F and G). Then O(k(F \Gamma G)) ' O(kF \Gamma G1 \Gamma \Delta \Delta \Delta \Gamma Gk) for arbitrary elements G1; : : : ; Gk of the linear system jGj. If we choose such elements Gj in general position, the lemma follows from the Riemann-Roch formula applied to the restriction morphism H 0 (X; O(kF )) ! L H 0 (Gj ; O(kF_G j ). \Lambda 17.2. Corollary. Let F and G be nef line bundles over X. If F is big and if m ? nF n\Gamma 1 \Delta G=F n , then O(mF \Gamma G) can be given a (possibly singular) Hermitian metric h, having a positive definite curvature form, i.e. such that \Theta h(mF \Gamma G) * ffl!; ffl ? 0, for a K"ahler metric !. Proof. In fact, if A is ample and ffl 2 Q+ is small enough, Lemma (17.1) implies that a certain multiple k(mF \Gamma G \Gamma fflA) admits a section. Let E be the divisor of this section and let ! = \Theta (A) 2 c1(A) be a K"ahler metric representing the curvature form of A. Then mF \Gamma G j fflA + 1 k E can be given a singular metric h with curvature form \Theta h(mF \Gamma G) = ffl\Theta (A) + 1 k [E] * ffl!. \Lambda We now consider the problem of obtaining a non-trivial section of mL. The idea of [Siu93] is to obtain a more general criterion for the ampleness of mL \Gamma B when B is nef. In this way, we will be able to subtract from mL any undesired multiple of KX that would be added to L, by application of the Nadel Vanishing Theorem (for this, we simply replace B, by B plus a multiple of KX + (n + 1)L). 17.3. Proposition. Let L be an ample line bundle on a projective manifold X of dimension n, and let B be an nef line bundle on X. Then KX + mL \Gamma B admits a non-zero section for an integer m satisfying m ^ n L n\Gamma 1 \Delta B L n + n + 1: 87 17. an effective version of matsusaka's big theorem Proof. Let m0 be the smaller integer ? n L n\Gamma 1 \Delta B Ln . Then m0L \Gamma B can be given a singular Hermitian metric h with positive definite curvature. By virtue of the Nadel vanishing theorem, we obtain H q (X; O(KX + mL \Gamma B) \Omega J (h)) = 0 for q * 1; therefore P (m) = h 0 (X; O(KX + mL \Gamma B) \Omega J (h)) is a polynomial for m * m0. Since P is a polynomial of degree n which is not identically zero, there exists an integer m 2 [m0; m0 + n] which is not a root. Therefore there exists a non-trivial section of H 0 (X; O(KX + mL \Gamma B)) oe H 0 (X; O(KX + mL \Gamma B) \Omega J (h)) for some m 2 [m0; m0 + n], as stated. \Lambda 17.4. Corollary. If L is ample and B is nef, then mL \Gamma B has a non-zero section for at least one integer m ^ n ` L n\Gamma 1 \Delta B + L n\Gamma 1 \Delta KX L n + n + 1 ' : Proof. According to the result of Fujita (16.4), KX + (n + 1)L is nef. We can therefore replace B by B + KX + (n + 1)L in Prop. (17.3). Corollary (17.4) follows. \Lambda 17.5. Remark. We do not know if the bound obtained in the above corollary is optimal, but it is certainly not very far from being it. Indeed, even for B = 0, the multiplicative factor n cannot be replaced by a number smaller than n=2. To see this, take for example for X a product C1 \Theta \Delta \Delta \Delta \Theta Cn of curves Cj of large enough genus gj, and L = O(a1[p1]) \Omega \Delta \Delta \Delta \Omega O(an[pn]); B = 0. Our sufficient condition so that jmLj 6= ; becomes in this case m ^ P (2gj \Gamma 2)=aj + n(n + 1), while for a generic choice of pj the bundle mL admits sections only if maj * gj for all j. The inaccuracy of our inequality thus plays more on one multiplicative factor 2 when a1 = \Delta \Delta \Delta = an = 1 and g1 AE g2 AE \Delta \Delta \Delta AE gn ! +1. In addition, the additive constant n + 1 is already the best possible when B = 0 and X = P n . \Lambda Up to this point, the method was not really sensitive to the presence of singularities (Lemma (17.1) is still true in the singular case as is easily seen by passing to a desingularization of X). In the same way, as we observed with remark (15.16), the Nadel vanishing theorem still remains essentially valid. Prop. (17.3) can then be generalized as follows: 17.6. Proposition. Let L be an ample line bundle on a projective manifold X of dimension n, and let B be an nef line bundle on X. For any (reduced) algebraic subvariety Y of X of dimension p, there exists an integer m ^ p L p\Gamma 1 \Delta B \Delta Y L p \Delta Y + p + 1 such that the sheaf !Y \Omega OY (mL \Gamma B) has a non-zero section. \Lambda By applying a suitable induction procedure relying on the results above, we can now improve the effective bound obtained by Siu [Siu93] for Matsusaka's big theorem. Our statement will depend on the choice of a constant *n such that j.-p. demailly, part ii: L 2 estimations and vanishing theorems 88 m(KX +(n+2)L)+G is very ample for m * *n and all nef line bundles G. Theorem (0.2 c) shows that *n ^ \Gamma 3n+1 n \Delta \Gamma 2n (a more elaborate argument concerning the recent results of Angehrn-Siu [AS94] allows us in fact to see that *n ^ n 3 \Gamma n 2 \Gamma n\Gamma 1 for n * 2). Of course, one expects with this that *n = 1 for all n, if one believes that the conjecture of Fujita is true. 17.7. Effective version of Matsusaka's Big Theorem. Let L and B be nef line bundles on a projective manifold X of dimension n. Assume that L is ample and let H = *n(KX + (n + 2)L). Then mL \Gamma B is very ample for m * (2n) (3 n\Gamma 1 \Gamma 1)=2 (L n\Gamma 1 \Delta (B + H)) (3 n\Gamma 1 +1)=2 (L n\Gamma 1 \Delta H) 3 n\Gamma 2 (n=2\Gamma 3=4)\Gamma 1=4 (L n ) 3n\Gamma 2(n=2\Gamma 1=4)+1=4 : In particular mL is very ample for m * Cn(L n ) 3 n\Gamma 2 ` n + 2 + L n\Gamma 1 \Delta KX L n '3n\Gamma 2(n=2+3=4)+1=4 with Cn = (2n) (3 n\Gamma 1 \Gamma 1)=2 (*n) 3 n\Gamma 2 (n=2+3=4)+1=4 . Proof. We utilize Th. (3.1) and Prop. (17.6) to construct by induction a sequence of algebraic subvarieties (not necessarily irreducible) X = Yn oe Yn\Gamma 1 oe \Delta \Delta \Delta oe Y2 oe Y1 such that Yp = [jYp;j is of dimension p, Yp\Gamma 1 being obtained for each p * 2 as the union of the set of zeros of the sections oep;j 2 H 0 (Yp;j; OY p;j (mp;jL \Gamma B)) for suitable integers mp;j * 1. We proceed by induction on the decreasing values of the dimension p, and we seek to obtain with each step an upper bound mp for the integer mp;j. By virtue of Cor. (17.4), we can find an integer mn such that mnL \Gamma B admits a non-trivial section oen for mn ^ n L n\Gamma 1 \Delta (B + KX + (n + 1)L) L n ^ n L n\Gamma 1 \Delta (B + H) L n : Now suppose that the sections oen; : : : ; oep+1;j have already been constructed. One then obtains by induction a p-cycle ~ Yp = P _p;jYp;j defined by ~ Yp = sum of the divisors of zeros of the sections oep+1;j on the components ~ Yp+1;j, where the multiplicity _p;j of Yp;j ae Yp+1;k is obtained by multiplying the corresponding multiplicity _p+1;k by the order of vanishing of oep+1;k along Yp;j. We obtain the equality of cohomology classes ~ Yp j X (mp+1;kL \Gamma B) \Delta (_p+1;kYp+1;k) ^ mp+1L \Delta ~ Yp+1: By induction, we then obtain the numerical inequality ~ Yp ^ mp+1 \Delta \Delta \Delta mnL n\Gamma p : Now, for each component Yp;j, Prop. (17.6) shows that there exists a section of !Y p;j \Omega OY p;j (mp;jL \Gamma B) for a certain integer mp;j ^ p L p\Gamma 1 \Delta B \Delta Yp;j L p \Delta Yp;j + p + 1 ^ pmp+1 \Delta \Delta \Delta mnL n\Gamma 1 \Delta B + p + 1: 89 17. an effective version of matsusaka's big theorem We have used here the obvious lower bound L p\Gamma 1 \Delta Yp;q * 1 (this bound is besides undoubtly one of weak points of the method...). The degree Yp;q by comparison to H admits the upper bound ffip;j := H p \Delta Yp;j ^ mp+1 \Delta \Delta \Delta mnH p \Delta L n\Gamma p : The Hovanski-Teissier concavity inequality gives (L n\Gamma p \Delta H p ) 1 p (Ln) 1\Gamma 1 p ^ Ln\Gamma 1 \Delta H ([Hov79], [Tei79, 82], also see [Dem93]), which makes it possible to express our bounds in terms of only the intersection numbers L n and L n\Gamma 1 \Delta H. We then obtain ffip;j ^ mp+1 \Delta \Delta \Delta mn (L n\Gamma 1 \Delta H) p (L n ) p\Gamma 1 : We have need of the following lemma, which will be proven shortly. 17.8. Lemma. Let H be a very ample line bundle on a projective algebraic manifold X, and let Y ae X be an irreducible algebraic subvariety of dimension p. If ffi = H p \Delta Y is the degree of Y with support in H, the sheaf Hom \Gamma !Y ; OY ((ffi \Gamma p \Gamma 2)H) \Delta has a non-trivial section. According to Lemma (17.8), there exists a non-trivial section of Hom \Gamma !Y p;j ; OY p;j ((ffip;j \Gamma p \Gamma 2)H) \Delta : By combining this section with the section of !Y p;j \Omega OY p;j (mp;jL \Gamma B) already constructed, we obtain a section of OY p;j (mp;jL \Gamma B + (ffip;j \Gamma p \Gamma 2)H) on Yp;j. We do not want H appearing at this stage, which is why we will replace B by B + (ffip;q \Gamma p \Gamma 2)H. We obtain then a section oep;j of OY p;j (mp;jL \Gamma B) for a certain integer mp;j such that mp;j ^ pmp+1 \Delta \Delta \Delta mnL n\Gamma 1 \Delta (B + (ffip;j \Gamma p \Gamma 2)H) + p + 1 ^ pmp+1 \Delta \Delta \Delta mnffip;jL n\Gamma 1 \Delta (B + H) ^ p(mp+1 \Delta \Delta \Delta mn) 2 (L n\Gamma 1 \Delta H) p (L n ) p\Gamma 1 L n\Gamma 1 \Delta (B + H): Consequently, by setting m = nL n\Gamma 1 \Delta (B + H), we obtain the descending inductive relation mp ^ M (L n\Gamma 1 \Delta H) p (L n ) p\Gamma 1 (mp+1 \Delta \Delta \Delta mn) 2 for 2 ^ p ^ n \Gamma 1; on the basis of the initial value mn ^ M=L n . Let (mp) be the sequence of numbers obtained by this inductive formula by replacing the respective inequalities by equalities. Thus we have mp ^ mp with mn\Gamma 1 = M 3 (L n\Gamma 1 \Delta H) n\Gamma 1 =(L n ) n and mp = L n L n\Gamma 1 \Delta H m 2 p+1 mp+1 for 2 ^ p ^ n \Gamma 2. Then by induction mp ^ mp = M 3 n\Gamma p (L n\Gamma 1 \Delta H) 3 n\Gamma p\Gamma 1 (n\Gamma 3=2)+1=2 (L n ) 3n\Gamma p\Gamma 1(n\Gamma 1=2)+1=2 : j.-p. demailly, part ii: L 2 estimations and vanishing theorems 90 We now show that m0L \Gamma B is nef for m0 = max(m2; m3; : : : ; mn; m2 \Delta \Delta \Delta mnL n\Gamma 1 \Delta B): Indeed, let C ae X be an arbitrary irreducible curve. Alternatively, C = Y1;j for a certain j, or else there exists an integer p = 2; : : : ; n such that C is contained in YpnYp\Gamma 1. If C ae Yp;j nYp\Gamma 1, then oep;j is not identically zero on C. Therefore (mp;jL \Gamma B)_C is of positive degree or zero and (m0L \Gamma B) \Delta C * (mp;jL \Gamma B) \Delta C * 0: In addition, if C = Y1;j, then (m0L \Gamma B) \Delta C * m0 \Gamma B \Delta ~ Y1 * m0 \Gamma m2 \Delta \Delta \Delta mnL n\Gamma 1 \Delta B * 0: According to the definition of *n (and the proof where such a constant exists, cf. (0.2c)), H + G is very ample for any nef line bundle G, in particular H + m0L \Gamma B is very ample. We again replace B by B + H. This substitution has the effect of replacing M by the new constant m = n(L n\Gamma 1 \Delta (B + 2H)) and m0 by m0 = max(mn; mn\Gamma 1; : : : ; m2; m2 \Delta \Delta \Delta mnL n\Gamma 1 \Delta (B + H)): The latter term being the largest estimation of mp implies m0^ M (3 n\Gamma 1 \Gamma 1)=2 (L n\Gamma 1 \Delta H) (3n\Gamma 2\Gamma 1)(n\Gamma 3=2)=2+(n\Gamma 2)=2 L n\Gamma 1 \Delta (B+H)) (Ln)(3 n\Gamma 2\Gamma 1)(n\Gamma 1=2)=2+(n\Gamma 2)=2+1 ^ (2n) (3 n\Gamma 1 \Gamma 1)=2 (L n\Gamma 1 \Delta (B+H)) (3n\Gamma 1+1)=2 (L n\Gamma 1 \Delta H) 3n\Gamma 2(n=2\Gamma 3=4)\Gamma 1=4 (Ln)3 n\Gamma 2(n=2\Gamma 1=4)+1=4 \Lambda Proof of lemma (17.8). Let X ae P N be the embedding given by H, so that H = OX (1). There exists a projective linear map P n ! P p+1 for which the restriction ss : Y ! P p+1 to Y is a finite and birational morphism of Y onto an algebraic hypersurface Y 0 of degree ffi in P p+1 . Let s 2 H 0 (P p+1 ; O(ffi)) be the polynomial of degree ffi defining Y 0 . We claim that for any small Stein open subset W ae P p+1 and any holomorphic p-form u, L 2 on Y 0 "W , there exists a holomorphic (p + 1)-form ~u, L 2 on W , with values in O(ffi), such that ~u_Y 0"W = u ^ ds. In fact, this is precisely the conclusion of the L 2 extension theorem of Ohsawa-Takegoshi [OT87], [Ohs88] (also see [Man93] for a more general version of this result). One can equally invoke standard arguments in local algebra (see Hartshorne [Har77], th. III-7.11). Since KPp+1 = O(\Gamma p \Gamma 2), the form ~u can be considered as a section of O(ffi \Gamma p \Gamma 2) on W , consequently the morphism of sheaves u 7! u ^ ds extends to a global section of Hom \Gamma !Y 0; OY 0(ffi \Gamma p \Gamma 2) \Delta . The inverse image of ss \Lambda furnishes a section of Hom \Gamma ss \Lambda !Y 0 ; OY ((ffi \Gamma p \Gamma 2)H) \Delta . Since ss is finite and generically 1 : 1, it is easy to see that ss \Lambda !Y 0 = !Y . The lemma follows. \Lambda 17.9. Remark. In the case of surfaces (n = 2), we can take *n = 1 according to the result of I. Reider [Rei88], and the arguments developed above ensure that mL is very ample for m * 4 (L \Delta (KX + 4L)) 2 L 2 : By working through the proof more carefully, it can be shown that the multiplicative factor 4 can be replaced by 2. In fact, Fernandez del Busto has recently shown that 91 17. an effective version of matsusaka's big theorem mL is very ample for m ? 1 2 ^ (L \Delta (KX + 4L) + 1) 2 L 2 + 3 * ; and an example of G. Xiao shows that this bound is essentially optimal (see [FdB94]). Matsusaka's big theorem yields a number of other important finiteness results. One of the prototypes of these results is the following statement. 17.10. Corollary. There exists only a finite number of families of deformations of polarized projective manifolds (X; L) of dimension n, where L is an ample line bundle for which the intersection numbers L n and KX \Delta L n\Gamma 1 are fixed. Proof. Indeed, since L n and KX \Delta L n\Gamma 1 are fixed, there in fact exists a calculable integer m0 such that m0L is very ample. We then obtain an embedding \Phi = \Phi jm 0Lj : X ! P N such that \Phi \Lambda O(1) = \Sigma m0L. The image Y = \Phi (X) is of degree deg(Y ) = Z Y c1 \Gamma O(1) \Delta n = Z X c1(\Sigma m0L) n = m n 0 L n : This implies that Y is a point of one of the components of the Chow scheme of algebraic subvarieties Y of a given dimension and degree in P N for which O(1)_Y is divisible by m0. More precisely a point of an open set corresponding to a nonsingular subvariety. 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[Wel80] R.O. Wells, Differential analysis on complex manifolds, Graduate Texts in Math. 65, 2nd edition, Springer-Verlag, Berlin (1980). Frobenius and Hodge Degeneration Luc Illusie Universit'e de Paris-Sud, D'epartment de Math 'matiques, B^atiment 425, 91405 Orsay Cedex, France 96 97 0. introduction In [D-I], the Hodge degeneration theorem and the Kodaira-Akizuki-Nakano vanishing theorem for smooth projective varieties over a field of characteristic zero are shown by methods of algebraic geometry in characteristic p ? 0. These present notes will serve as an introduction to the subject, with the intention of keeping the non-specialist in mind (who will be able to also consult the presentation of Oesterl'e [O]). Thus we will assume known by the reader only some rudiments of the theory of schemes (EGA I 1-4, [H2] II 2-3). On the other hand, we require of the reader a certain familiarity with homological algebra. The results of [D-I] are expressed simply in the language of derived categories. Although it is possible to avoid there the recourse, see for example [E-V], we prefer to place it in its context, which appears more natural. However, to help the beginner, we recall in n ffi 4 the basic definitions and some essential points. 0. Introduction Let X be a complex analytic manifold. By the Poincar'e Lemma, the de Rham complex \Omega ffl X of holomorphic forms on X is a resolution of the constant sheaf C . As a result, the augmentation C ! \Omega ffl X defines an isomorphism (for all n) (0.1) H n (X; C ) , \Gamma ! H n DR (X) = H n (X; \Omega ffl X ); where the second term, called the de Rham cohomology of X (in degree n), is the n-th hypercohomology group of X with values in \Omega ffl X . The first spectral sequence of hypercohomology abuts to the de Rham cohomology of X (0.2) E p;q 1 = H q (X; \Omega p X ) ) H p+q DR (X); which is called the Hodge to de Rham spectral sequence (or Hodge-Fr"olicher) (cf. [De] n ffi 9). Let us assume X is compact. Then, by the finiteness theorem of CartanSerre, the H q (X; \Omega p X ), and therefore all the terms of the spectral sequence (0.2) are finite dimensional C -vector spaces. If we set bn = dim H n DR (X) = dim H n (X; C ) (n-th Betti number of X) and h p;q = dim H q (X; \Omega p X ) (Hodge number), we have (0.3) bn ^ X p+q=n h pq ; with equality for all n if and only if (0.2) degenerates at E1. Suppose in addition that X is K"ahler. Then by Hodge theory, the Hodge spectral sequence of X degenerates at E1 : this is the Hodge degeneration theorem ([De] 9.9). Denote by 0 = F n+1 ae F n ae \Delta \Delta \Delta ae F p = F p H n DR (X) ae \Delta \Delta \Delta ae F 0 = H n DR (X) the resulting filtration of the Hodge spectral sequence (Hodge filtration). By degeneration, one has a canonical isomorphism (0.4) E p;q 1 = H q (X; \Omega p X ) ' E pq 1 = F p =F p+1 : We put H p;q = F p " F q ; where the bar denotes complex conjugation on H n DR (X), defined by means of (0.1), and the isomorphism H n (X; C ) ' H n (X; R) \Omega C . It follows that H p;q = H q;p : luc illusie, frobenius and hodge degeneration 98 Further, Hodge theory furnishes the following results ([De] 9.10): (a) the composite homomorphism H p;q ,! F p H p+q DR (X) i F p =F p+1 is an isomorphism (i.e. H p;q is a complement of F p+1 in F p ); whence, by composing with (0.4), determines an isomorphism (0.5) H p;q ' H q (X; \Omega p X ); (b) one has, for all n, (0.6) H n DR (X) = M p+q=n H p;q ; (Hodge decomposition). These results apply in particular to the complex analytic manifold X associated to a smooth projective scheme X over C . The difference between (a) and (b), which is of a transcendental nature, utilizes complex conjugation in an essential way. The Hodge degeneration can in this case be formulated in a purely algebraic manner. The de Rham complex of X is indeed the complex of analytic sheaves associated to the algebraic de Rham complex \Omega ffl X of X over C (a complex of sheaves in the Zariski topology, for which the components are locally free coherent sheaves). The canonical morphism (of ringed spaces) X ! X induces homomorphisms on the Hodge and de Rham cohomologies H q (X; \Omega p X ) ! H q (X; \Omega p X );(0.7) H n DR (X) ! H n DR (X);(0.8) where H n DR (X) = H n (X; \Omega ffl X ). We make use of the Hodge to algebraic de Rham spectral sequence (0.9) E p;q 1 = H q (X; \Omega p X ) ) H p+q DR (X); and a morphism of (0.9) in (0.2) inducing (0.7) and (0.8) respectively on the initial terms and the abutment. By the comparison theorem of Serre [GAGA], (0.7) is an isomorphism, and therefore the same holds for (0.8). Consequently, the degeneration at E1 of (0.2) is equivalent to that of (0.9). In other words, if one sets h p;q (X) = dim H q (X; \Omega p X ); h n (X) = dim H n DR (X); the Hodge degeneration theorem for X is expressed by the (purely algebraic) relation (0.10) h n (X) = X p+q=n h p;q (X): More generally, if X is a smooth and proper scheme over a field k, one can consider the de Rham complex \Omega ffl X=k of X over k, and one still has a Hodge to de Rham spectral sequence (0.11) E pq 1 = H q (X; \Omega p X=k ) ) H p+q DR (X=k) (where H n DR (X=k) = H n (X; \Omega ffl X=k )), formed of finite-dimensional k-vector spaces. If k is of characteristic zero, the Hodge degeneration theorem implies the degeneration of (0.11) at E1 : standard techniques (cf. n ffi 6) indeed make it possible to go back initially to k = C , then with the aid of Chow's Lemma and of the resolution of singularities one reduces the proper case to the projective case ([D0]). There 99 1. schemes: differentials, the de rham complex are those who have long sought for a purely algebraic proof of the degeneration of (0.11) at E1 for k of characteristic zero. Faltings [Fa1] was the first to give a proof of it independent of Hodge theory 2 . A simplification of crystalline techniques due to Ogus [Og1], Fontaine-Messing [F-M] and Kato [Ka1] led, shortly thereafter, to the elementary proof presented in [D-I]. We refer to the introduction of [D-I] and to [O] for a broad overview. We only indicate that the degeneration of (0.11) (for k of characteristic zero) is proven by reduction to the case where k is of characteristic p ? 0, where, however, it can happen that the degeneration is automatic! This proof is based however on the help of some additional hypothesis on X (upper bound of the dimension, liftability) which is sufficient for our purposes (see 5.6 for a precise statement). We explain in n ffi 6 the well-known technique which allows us to go from characteristic p ? 0 to characteristic zero. The degeneration theorem in characteristic p ? 0 to which we have just alluded follows from a decomposition theorem (5.1), relying on some classical properties of differential calculus in characteristic p ? 0 (Frobenius endomorphism and Cartier isomorphism), which we recall in n ffi 3, after having summarized, in n ffi 1 and 2, the formalism of differentials and smoothness on schemes. The aforementioned decomposition theorem furnishes at the same time an algebraic proof of the Kodaira-Akizuki-Nakano vanishing theorem for the smooth projective varieties over a field of characteristic zero (6.10 and [De] 11.7). The last two sections are of a more technical nature: We outline the evolution of the subject since the publication of [D-I], and, in the appendix, we describe some complementary results due to Mehta-Srinivas [Me-Sr] and Nakkajima [Na]. 1. Schemes: differentials, the de Rham complex We recall here the definition and basic properties of differential calculus over schemes. The reader will find a complete treatment in (EGA IV 16.1-16.6); also see [B-L-R] 2.1 and [H2] II 8 for an introduction. 1.1. We say that a morphism of schemes i : T0 ! T is a thickening of order 1 (or by abuse, that T is a thickening of order 1 of T0) if i is a closed immersion defined by an ideal of OT of square zero. If T and T0 are affine, with rings A and A0, such a morphism corresponds to a surjective homomorphism A ! A0 for which the kernel is an ideal of square zero. The schemes T and T0 have the same underlying space, and the ideal a of i, annihilated by a, is a quasi-coherent OT 0 ( = OT =a)-module. Let j : X ! Z be an immersion, with ideal I (by definition, j is an isomorphism of X onto a closed subscheme j(X) of a larger open subset U of Z, and I is the quasi-coherent sheaf of ideals of U defining j(X) in U , (EGA I 4.1, 4.2)). Let Z1 be the subscheme 3 of Z, with the same underlying space as X, defined by the ideal I 2 . Then j factors (in a unique way) into X j1 \Gamma ! Z1 h1 \Gamma ! Z 2 The purists observe that this proof, which rests on the existence of the Hodge-Tate decomposition for p-adic 'etale cohomology of a smooth and proper variety over a local field of unequal characteristic, is not entirely "algebraic", in the sense of where it uses the comparison theorem of Artin-Grothendieck between 'etale cohomology and Betti cohomology for smooth and proper varieties over C . 3 At the expense of some abuse of notation, we will allow ourselves the flexibility of interchanging "immersion" (resp. "closed immersion") and "subscheme" (resp. "closed subscheme"); that amounts here to neglecting the isomorphism of X onto j(X). luc illusie, frobenius and hodge degeneration 100 where h1 is an immersion, and j1 is a thickening of order 1, with ideal I=I 2 ; one says that (j1; h1), or more simply Z1, is the first infinitesimal neighbourhood of j (or of X in Z). The ideal I=I 2 (which is a quasi-coherent OX-module) is called the conormal sheaf of j (or of X in Z). We denote it by NX=Z . 1.2. Let f : X ! Y be a morphism of schemes, and let \Delta : X ! Z := X \Theta Y X be the diagonal morphism. This is an immersion (closed if and only if X is separated over Y ) (EGA I 5.3). The conormal sheaf of \Delta is called the sheaf of K"ahler 1- differentials of f (or of X over Y ) and is denote by \Omega 1 X=Y ; we sometimes write \Omega 1 X=A instead of \Omega 1 X=Y if Y is affine with ring A. Thus we have a quasi-coherent OX -module, defined by (1.2.1) \Omega 1 X=Y = I=I 2 ; where I is the ideal of \Delta . Let X \Delta 1 \Gamma \Gamma ! Z1 ! Z be the first infinitesimal neighbourhood of \Delta . The two projections of Z = X \Theta Y X on X induce, by composition with Z1 ! Z, two Y -morphisms p1; p2 : Z1 ! X, which retract \Delta 1. The sheaf of rings of the scheme Z1, which has the same underlying space as X, is called the sheaf of principal parts of order 1 of X over Y , and is denoted by P 1 X=Y . We have, by construction, an exact sequence of abelian sheaves (1.2.2) 0 ! \Omega 1 X=Y ! P 1 X=Y ! OX ! 0; split by each of the ring homomorphisms j1; j2 : OX ! P 1 X=Y induced from p1; p2. The difference j2 \Gamma j1 is a homomorphism of abelian sheaves of OX in \Omega 1 X=Y , which is called the differential, and which is denoted by (1.2.3) dX=Y (or d) : OX ! \Omega 1 X=Y : If M is an OX -module, a Y -derivation of OX in M is any homomorphism of sheaves of f \Gamma 1 (OY )-modules D : OX ! M (where f \Gamma 1 denotes the inverse image functor for abelian sheaves) such that D(ab) = aDb + bDa for all local sections a; b of OX . We denote by DerY (OX ; M ), the set of Y derivations of OX in M , which is in a natural way an abelian group. The differential dX=Y is a Y -derivation of OX in \Omega 1 X=Y . One shows that it is universal, in the sense that for any Y -derivation D of OX in an OX -module M (not necessarily quasicoherent), there exists a unique homomorphism of OX -modules u : \Omega 1 X=Y ! M such that u ffi dX=Y = D, i.e. the homomorphism (1.2.4) Hom(\Omega 1 X=Y ; M ) ! DerY (OX ; M ); u 7! u ffi dX=Y is an isomorphism. The sheaf Hom(\Omega 1 X=Y ; OX ) is called the tangent sheaf of f (or of X over Y ), and is denoted by (1.2.5) TX=Y (or sometimes \Theta X=Y ). For any open subset U of X, (1.2.4) gives an isomorphism \Gamma (U; TX=Y ) ' DerY (OU ; OU ). Recall that one calls a Y -point of X a Y -morphism T ! X. By definition, X \Theta Y X "parameterizes" the set of pairs of 101 1. schemes: differentials, the de rham complex Y -points of X (i.e. represents the corresponding functor on the category of Y schemes). The geometric significance of the first infinitesimal neighbourhood Z1 of the diagonal of X over Y is that it parameterizes the pairs of Y -points of X neighbouring of order 1 (i.e. congruent modulo an ideal of square zero): More precisely, if i : T0 ! T is a thickening of order 1, with ideal a, where T is a Y -scheme, and if t1; t2 : T ! X are two Y -points of X which coincide modulo a (i.e. such that t1i = t2i = t0 : T0 ! X), then there exists a unique Y -morphism h : T ! Z1 such that p1h = t1 and p2h = t2. Moreover, if t \Lambda 1 ; t \Lambda 2 : OX ! t0\Lambda OT 4 are the homomorphisms of sheaves of rings associated to t1 and t2, t \Lambda 2 \Gamma t \Lambda 1 is a Y -derivation of X with values in t0\Lambda a, such that (1.2.6) (t \Lambda 2 \Gamma t \Lambda 1 )(s) = h \Lambda (ds) for any local section s of OX, where h \Lambda : \Omega 1 X=Y ! t \Lambda 0 a is the homomorphism of OXmodules induced by h (on the corresponding conormal sheaves of X in Z1 and T0 in T ). If f is a morphism of affine schemes, corresponding to a ring homomorphism A ! B, then Z = Spec B \Omega A B; \Delta corresponds to the ring homomorphism sending b1 \Omega b2 onto b1b2, with kernel J = \Gamma (Z; I). We have \Gamma (X; P 1 X=Y ) = (B \Omega A B)=J 2 , and we set (1.2.7) \Gamma (X; \Omega 1 X=Y ) = \Omega 1 B=A : The B-module \Omega 1 B=A = J=J 2 , for which the associated quasi-coherent sheaf is \Omega 1 X=Y , is called the module of K"ahler 1-differentials of B over A. The map d = dB=A = \Gamma (X; dX=Y ) : B ! \Omega 1 B=A is an A-derivation, satisfying a universal property that we leave to the reader to formulate. The homomorphisms j1; j2 : B ! (B \Omega A B)=J 2 of 1.1 are given by j1b = class of b \Omega 1; j2b = class of 1 \Omega b. Since J is generated by 1 \Omega b \Gamma b \Omega 1; \Omega 1 B=A is generated, as a B-module, by the image of d. It follows from this that if f is any given morphism of schemes, \Omega 1 X=Y is generated, as an OX -module, by the image of d. 1.3. Any commutative square (1.3.1) X 0 g \Gamma ! X f 0 # # f Y 0 h \Gamma ! Y defines in a canonical way, a homomorphism of OX0-modules (1.3.2) g \Lambda \Omega 1 X=Y ! \Omega 1 X0=Y 0 ; which sends 1 \Omega g \Gamma 1 (dX=Y s) onto dX0=Y 0 (1 \Omega g \Gamma 1 (s)). (If E is an OX -module, by definition g \Lambda E = OX0\Omega g\Gamma 1(O X) g \Gamma 1 (E).) This is an isomorphism if the square (1.3.1) is cartesian, i.e. if the morphism X 0 ! Y 0 \Theta Y X is an isomorphism. Moreover, in this case, the canonical homomorphism (1.3.3) f 0\Lambda \Omega 1 Y 0=Y \Phi g \Lambda \Omega 1 X=Y ! \Omega 1 X0=Y is an isomorphism. 4 Recall that T and T0 have the same underlying space. luc illusie, frobenius and hodge degeneration 102 1.4. Let X f \Gamma ! Y g \Gamma ! S be morphisms of schemes. Then the canonical sequence of homomorphisms (1.4.1) f \Lambda \Omega 1 Y=S ! \Omega 1 X=S ! \Omega 1 X=Y ! 0 is exact. 1.5. Let X i \Gamma ! Z f # . g Y be a commutative triangle, where i is an immersion, with ideal I. The differential dZ=Y induces a homomorphism d : NX=Z ! i \Lambda \Omega 1 Z=Y , and the sequence (1.5.1) NX=Z ! i \Lambda \Omega 1 Z=Y ! \Omega 1 X=Y ! 0 is exact. 1.6. Let X = A n Y = Y [T1; : : : ; Tn] be the affine space of dimension n over Y . The OX-module \Omega 1 X=Y is free, with basis dTi (1 ^ i ^ n). If Y is affine, with ring A, and if s 2 A[T1; : : : ; Tn], then ds = P (@s=@Ti)dTi, where the @s=@Ti are the usual partial derivatives. Properties 1.3 to 1.6, for which the verification is completely standard, are fundamental. It is by virtue of these that we can "calculate" the modules of differentials. For more details, see the indicated references above. 1.7. Let f : X ! Y be a morphism of schemes. For i 2 N, we denote by \Omega i X=Y = \Lambda i \Omega 1 X=Y the i-th exterior product of the OX -module \Omega 1 X=Y . (It is agreed that \Omega 0 X=Y = OX .) One shows that there exists a unique family of maps d : \Omega i X=Y ! \Omega i+1 X=Y satisfying the following conditions: (a) d is a Y -anti-derivation of the exterior algebra L \Omega i X=Y , i.e. d is f \Gamma 1 (OY )-linear and d(ab) = da ^ b + (\Gamma 1) i a ^ db for a homogenous of degree i, (b) d 2 = 0, (c) da = dX=Y (a) for a of degree zero. The corresponding complex is called the de Rham complex of X over Y and is denoted by (1.7.1) \Omega ffl X=Y (or \Omega ffl X=A if Y is affine with ring A). It depends functorially on f : A square (1.3.1) gives a homomorphism of complexes (which is also a homomorphism of algebras) (1.7.2) \Omega ffl X=Y ! g\Lambda \Omega ffl X0=Y 0 : However, one must be aware that even if for each i, the homomorphism \Omega i X=Y ! g\Lambda \Omega i X0=Y 0 is the adjoint of a homomorphism g \Lambda \Omega i X=Y ! \Omega i X0=Y 0 , one cannot in 103 smoothness and liftings general define a complex g \Lambda \Omega ffl X=Y for which the differential is a Y 0 -anti-derivation compatible with that of \Omega ffl X0=Y 0 . 2. Smoothness and liftings There are a number of ways of presenting the theory of smooth morphisms. We follow (or rather, summarize) here the presentation of EGA, where smoothness is defined by the existence of infinitesimal liftings (EGA IV 17). In addition to its elegance, this definition has the advantage of transposing itself to other contexts, for example that of the geometric logarithm (cf. [I6]). Other points of view are adopted in (SGA 1 II and III), where the emphasis is placed on the notion of an 'etale morphism, and [B-L-R] 2.2, where this is the jacobian criterion (cf. 2.8), which is taken as the starting point. 2.1. Let f : X ! Y be a morphism of schemes. We say that f is locally of finite type (resp. locally of finite presentation) if, for any point x of X, there exists an affine open neighbourhood U of x and an affine open neighbourhood V of y = f (x) such that f (U ) ae V and that the homomorphism of rings A ! B associated to U ! V makes B an A-algebra of finite type (i.e. a quotient of an algebra of polynomials A[t1; : : : ; tn]) (resp. of finite presentation (i.e. a quotient of an algebra of polynomials A[t1; : : : ; tn] by an ideal of finite type)). If Y is locally Noetherian, "locally of finite type" is equivalent to "locally of finite presentation", and if it is, then it follows that X is locally Noetherian. If f : X ! Y is locally of finite presentation, the OX -module \Omega i X=Y is of finite type for all i, therefore coherent if Y is locally Noetherian. 2.2. Let f : X ! Y be a morphism of schemes. We say that f is smooth (resp. net (or non-ramified), resp. 'etale) if f is locally of finite presentation and if the following condition is satisfied: For any commutative diagram (2.2.1) X g0 % # f T0 i \Gamma ! T ! Y where i is a thickening of order 1 (1.1), there exists, locally in the Zariski topology on T , a (resp. at most one, resp. a unique) Y -morphism g : T ! X such that gi = g0. It follows immediately from the definition that the composite of two smooth morphisms (resp. net, resp. 'etale) is smooth (resp. net, resp. 'etale), and that if f : X ! Y is smooth (resp. net, resp. 'etale), it is the same with the morphism f 0 : X 0 ! Y 0 induced by a base change Y 0 ! Y . If for i = 1; 2; fi : Xi ! Y is smooth (resp. net, resp. 'etale), the fiber product f = f1 \Theta Y f2 : X1 \Theta Y X2 ! Y is therefore smooth (resp. net, resp. 'etale). Additionally it is immediate that the projection of the affine line A 1 Y = Y [t] ! Y is smooth, and it is therefore the same for the projection of the space A n Y ! Y . Remarks 2.3. (a) Because of the uniqueness which allows a gluing together, we can omit in the definition of 'etale, locally in the Zariski topology. On the other hand, we cannot do it in the definition of smooth. There exist a cohomological obstruction that we will later specify, to the existence of a global extension g of g0. luc illusie, frobenius and hodge degeneration 104 (b) If n is an integer * 1, we say that a morphism of schemes i : T0 ! T is a thickening of order n if i is a closed immersion defined by an ideal I such that I n+1 = 0. If Tm denotes the closed subscheme of T defined by I m+1 , i itself factors into a sequence of thickenings of order 1 : T0 ! T1 ! \Delta \Delta \Delta ! Tm ! Tm+1 ! \Delta \Delta \Delta ! Tn: In Definition 2.2, we can therefore replace thickening of order 1 by thickening of order n. The following proposition summarizes the essential properties of differentials associated to smooth morphisms (resp. net, resp. 'etale). Proposition 2.4. (a) If f : X ! Y is smooth (resp. net), the OX -module \Omega 1 X=Y is locally free of finite type (resp. zero). (b) In the situation of 1.4, if f is smooth, the sequence (1.4.1) extended by a zero to the left (2.4.1) 0 ! f \Lambda \Omega 1 Y=S ! \Omega 1 X=S ! \Omega 1 X=Y ! 0 is exact and locally split. In particular, if f is 'etale, the canonical homomorphism f \Lambda \Omega 1 Y=S ! \Omega 1 X=S is an isomorphism. (c) In the situation of 1.5, if f is smooth, the sequence (1.5.1) extended by a zero to the left (2.4.2) 0 ! NX=Z ! i \Lambda \Omega 1 Z=Y ! \Omega 1 X=Y ! 0 is exact and locally split. In particular, if f is 'etale, the canonical homomorphism NX=Z ! i \Lambda \Omega 1 Z=Y is an isomorphism. 2.5. The verification of 2.4 is not difficult (EGA IV 17.2.3), but unfortunately somewhat scattered in (EGA 0IV 20). Here is an outline. The key ingredient is the following. If f : X ! Y is a morphism of schemes and I a quasi-coherent OX -module, we call a Y -extension of X by I, a Y -morphism i : X ! X 0 which is a thickening of order 1 with ideal I. Two Y -extensions i1 : X ! X1 and i2 : X ! X2 of X by I are said to be equivalent if there exists a Y -isomorphism g of X1 onto X2 such that gi1 = i2 and that g induces the identity on I. An analogous construction to this is the "Baer sum" for extensions of modules over a ring associated to the set ExtY (X; I) of equivalence classes of Y -extensions of X by I with a structure of an abelian group, with neutral element the trivial extension defined by the algebra of dual numbers OX \Phi I. Assertion (c) follows immediately from the definition: The smoothness of f indeed implies that the first infinitesimal neighbourhood i1 of i retracts locally onto X, and the choice of a retraction r permits the splitting (2.4.2) (by the derivation associated to IdZ 1 \Gamma i1 ffi r, cf. (1.2.6)). Assume f is smooth. If I is a quasi-coherent OX -module and if i : X ! Z is a Y -extension of X by I, the sequence (2.4.2) is therefore an extension of OX -modules e(i) of \Omega 1 X=Y by I. One can show that i 7! e(i) gives an isomorphism (2.5.1) ExtY (X; I) ! Ext 1 OX (\Omega 1 X=Y ; I) 105 smoothness and liftings (cf. [I1] I, chap. II, 1.1.9. We define an inverse of (2.5.1) by associating to an extension M of \Omega 1 X=Y by I, the Y -extension Z of X defined in the following way: Identify, via j1, the sheaf of principal parts P 1 X=Y (1.2.2) with the ring of dual numbers OX \Phi \Omega 1 X=Y , and denote by F = OX \Phi M the ring of dual numbers over M ; the extension M makes F an f \Gamma 1 (OY )-extension of P 1 X=Y by I. That is, if E = F \Theta P1 X=Y OX is the "pull-back" of F by the homomorphism j2 = j1 + dX=Y : OX ! P 1 X=Y , then E is a f \Gamma 1 (OY )-extension of OX by I, which defines the Y extension Z). Since f is smooth, any Y -extension of X by I is locally trivial, and therefore by virtue of (2.5.1), it follows from this that the sheaf Ext 1 OX (\Omega 1 X=Y ; I) (associated to the presheaf U 7! Ext 1 OU (\Omega 1 U=Y ; IjU )) is zero, and therefore also that Ext 1 OU (\Omega 1 U=Y ; J) = 0 for all open subsets U of X and all quasi-coherent OU -modules J. Since \Omega 1 X=Y is of finite type (2.1), it follows that \Omega 1 X=Y is locally free of finite type, which proves the part of (a) relative to the smooth case. (The relative part of the net case is immediate: For any Y -scheme X, if i : X ! Z is the trivial Y -extension of X by a quasi-coherent OX -module I, the set of Y -retractions of Z on X is identified with Hom(\Omega 1 X=Y ; I) by r 7! r \Gamma r0, where r0 corresponds to the natural injection of OX in OX \Phi I, cf. (1.2.6).) In particular, it follows from (a) and (2.5.1) that if X is an affine scheme and is smooth over Y , we have ExtY (X; I) = 0 for any quasi-coherent OX -module I. Finally, we arrive at (b), by using, for X; Y; S affine, and any given f , the natural exact sequence (EGA 0IV 20.2.3) (2.5.2) 0 ! DerY (OX; I) ! DerS(OX ; I) ! DerS(OY ; f\Lambda I) ! @ \Gamma ! ExtY (X; I) ! ExtS(X; I) ! ExtS(Y; f\Lambda I); where the arrows other than @ are the obvious arrows of functoriality, and @ associates to an S-derivation D : OY ! f\Lambda I the Y -extension defined by the ring of dual numbers OX \Phi I and the homomorphism a 7! f \Lambda a + Da of OY in f\Lambda (OX \Phi I). Observe that if f : X ! Y is a morphism locally of finite presentation of affine schemes (i.e. corresponding to a homomorphism of rings A ! B making B an A-algebra of finite presentation), then, for that f is smooth, it is necessary and sufficient that for any quasi-coherent OX -module I, we have ExtY (X; I) = 0 (the sufficiency rises from the definition, and the necessity was already noted above). Assertions 2.4 (b) and (c) have converses, which furnish a very convenient criteria of smoothness. Their verfication is easy, starting from previous considerations. Proposition 2.6. (a) In the situation of 1.4, assume gf smooth. If the sequence (2.4.1) is exact and locally split, then f is smooth. If the canonical homomorphism f \Lambda \Omega 1 Y=S ! \Omega 1 X=S is an isomorphism, then f is 'etale. (b) In the situation of 1.5, assume g smooth. If the sequence (2.4.2) is exact and locally split, then f is smooth. If the canonical homomorphism NX=Z ! i \Lambda \Omega 1 Z=Y is an isomorphism, then f is 'etale. 2.7. Let f : X ! Y be a smooth morphism, assume given x a point of X, and denote by k(x) the residue field of the local ring OX;x. Let s1; : : : ; sn be sections of OX in a neighbourhood of x for which the differentials form a basis of \Omega 1 X=Y at x, i.e., chosen such that the images (dsi)x of dsi in \Omega 1 X=Y;x form a basis of this module over OX;x, or such that the images (dsi)x of dsi in \Omega 1 X=Y \Omega k(x) form a basis of luc illusie, frobenius and hodge degeneration 106 this vector space over k(x). Since \Omega 1 X=Y is locally free of finite type, there exists an open neighbourhood U of x such that the si are defined over U and that the dsi form a basis of \Omega 1 X=Y jU . The sj then define a Y -morphism of U in the affine space of dimension n over Y : s = (s1; : : : ; sn) : U ! A n Y = Y [t1; : : : ; tn]: According to 1.6 and 2.6 (a), s is 'etale. We say that the si form a local coordinate system of X on Y over U (or, if U is not specified, at x). A smooth morphism is therefore locally composed of an 'etale morphism and of the projection of a standard affine space. 2.8. Now assume given the situation of 1.5, by assuming g is smooth, and let x be a point of X. According to 2.4 (c) and 2.6 (b), for that f to be smooth in a neighbourhood of x, it is necessary and sufficient that there exists sections s1; : : : ; sr of I in a neighbourhood of x, generating Ix and such that the (dsi)(x) are linearly independent in \Omega 1 Z=Y (x) = \Omega 1 Z=Y \Omega k(x) (where k(x) is the residue field of OZ;x, which is also that of OX;x). For this reason, 2.6 (b) is referred to as the jacobian criterion. Suppose f is smooth in a neighbourhood of x (or at x, like one says sometimes), and let s1; : : : ; sr be sections of I generating I in a neighbourhood of x. Then, for that the si defines a minimal system of generators of Ix (i.e. induces a basis of I \Omega k(x) = Ix=mxIx, or still forms a basis of I=I 2 = NX=Z in a neighbourhood of x), it is necessary and sufficient that the (dsi)(x) are linearly independent in \Omega 1 Z=Y (x) 5 . Therefore, wherever this is the case, if we supplement the si by sections sj (r +1 ^ j ^ r +n) of OZ in a neighbourhood of x such that the (dsi)(x) (1 ^ i ^ r + n) form a basis of \Omega 1 Z=Y (x), then the si (1 ^ i ^ n) define an 'etale Y -morphism s from an open neighbourhood U of x in Z into the affine space A n+r Y , such that U " X is the inverse image of the linear subspace with equations t1 = \Delta \Delta \Delta = tr = 0: U " X ! U # # f A n Y ! A n+r Y In algebraic geometry, this statement plays the role of the implicit function theorem. 2.9. Let k be a field and let f : X ! Y = Spec k be a morphism. Assuming f smooth, then X is regular (i.e. for any point x of X, the local ring OX;x is regular, i.e. its maximal ideal mx can be generated by a regular sequence of parameters); moreover, if x is a closed point, k(x) is a finite separable extension of k, and the dimension of OX;x is equal to the dimension dimx X of the irreducible component of X containing x and of the rank of \Omega 1 X=Y at x. Conversely, if k is perfect, and if X is regular, then f is smooth. More generally, we have the following criterion, left as an easy verification from 2.7 and 2.8 : 5 Or still that the sequence (si) is OZ-regular at x, i.e. that the corresponding Kozul complex is a resolution of OX in a neighbourhood of x (cf. (SGA 6 VII 1.4) and (EGA IV 17.12.1)). 107 smoothness and liftings Proposition 2.10. Let f : X ! Y be a morphism locally of finite presentation (2.1). The following conditions are equivalent : (i) f is smooth; (ii) f is flat and the geometric fibers of f are regular schemes. (We say that f is flat if for any point x of X, OX;x is a flat module over OY;f(x). A geometric fiber of f is the reduced scheme of a fiber Xy = X \Theta Y Spec k(y) of f at a point y by an extension of scalars to an algebraic closure of k(y).) If f : X ! Y is smooth, and x is a point of X, the integer dimx(f ) := dimk(x) \Omega 1 X=Y \Omega k(x) = rgO X;x \Omega 1 X=Y;x is called the relative dimension of f at x. By the classical theory of dimension (EGA IV 17.10.2), this is the dimension of the irreducible component of the fiber Xf(x) containing x. Since \Omega 1 X=Y is locally free of finite type, it is a locally constant function of x. It is zero if and only if f is 'etale, in other words, f is 'etale if and only if f is locally of finite presentation, flat and net (it is this criterion which is taken as the definition of an 'etale in (SGA 1 I)). If f is smooth and of pure relative dimension r, i.e. of constant relative dimension equal to the integer r, then the de Rham complex \Omega ffl X=Y (1.7.1) is zero in degree ? r, and \Omega i X=Y is locally free of rank \Gamma r i \Delta ; in particular, \Omega r X=Y is an invertible OX -module. Smooth morphisms occupy a central place in the theory of infinitesimal deformations. The following two propositions summarize this. They are however of a more technical nature than the preceeding statements, and as they will be useful only in the proof of 5.1, we will advise the reader to refer to it at that time there. Proposition 2.11. Assume given a diagram (2.2.1), with f smooth. Let I be the ideal of i. (a) There exists an obstruction c(g0) 2 Ext 1 (g \Lambda 0 \Omega 1 X=Y ; I) for which the vanishing is necessary and sufficient for the existence of a Y morphism (global) g : T ! X extending g0 (i.e. such that gi = g0). (b) If c(g0) = 0, the set of extensions g of g0 is an affine space under Hom(g \Lambda 0 \Omega 1 X=Y ; I). Since \Omega 1 X=Y is locally free of finite type, there is a canonical isomorphism (2.11.1) Ext 1 (g \Lambda 0 \Omega 1 X=Y ; I) ' H 1 (T0; Hom(g \Lambda 0 \Omega 1 X=Y ; I)) (and Hom(g \Lambda 0 \Omega 1 X=Y ; I) ' g \Lambda 0 TX=Y \Omega I, where TX=Y is the tangent sheaf (1.2.5)). Set G = Hom(g \Lambda 0 \Omega 1 X=Y ; I). According to (1.2.6), if U is an open subscheme of T with corresponding U0 over T0, two extensions of g0jU 0 to U "differ" by a section of G over U0 (and being given an extension, one can modify it by "adding" a section of G). Since g0 locally extends by definition of the smoothness of f , we then conclude that the sheaf P over T0 associating to U0 the set of extensions of g0jU 0 to U , is a torsor under G. Assertions (a) and (b) follow from this: c(g0) is the class of this torsor. More explicitly, if (Ui)i2E is an open covering of T and gi an extension of g0 over Ui, then, over Ui " Uj, gi \Gamma gj is a Y -derivation Dij of OX with values in luc illusie, frobenius and hodge degeneration 108 g0 \Lambda (IjU i"Uj ), i.e. a homomorphism of \Omega 1 X=Y into g0 \Lambda (IjU i"Uj ), i.e. finally a section of G over Ui " Uj, and the (gij) form a cocycle, for which the class is c(g0). Note that if T (or what amounts to the same T0) is affine, then H 1 (T0; Hom(g \Lambda 0 \Omega 1 X=Y ; I)) = 0 and consequently g0 admits a global extension to T . Proposition 2.12. Assume given i : Y0 ! Y a thickening of order 1 with ideal I, and f0 : X0 ! Y0 a smooth morphism. (a) There exists an obstruction !(f0) 2 Ext 2 (\Omega 1 X0=Y0 ; f \Lambda 0 I) for which the vanishing is necessary and sufficient for the existence of a smooth lifting X0 over Y , i.e. by definition, of a smooth Y -scheme X equipped with a Y0-isomorphism Y0 \Theta Y X ' X0 6 . (b) If !(f0) = 0, the set of isomorphism classes of liftings of X0 over Y is an affine space under Ext 1 (\Omega 1 X0=Y0 ; f \Lambda 0 I) (where by definition, if X1 and X2 are liftings of X0, an isomorphism of X1 onto X2 is a Y -isomorphism of X1 on X2 inducing the identity on X0). (c) If X is a lifting of X0 over Y , the group of automorphisms of X (i.e. Y automorphisms of X inducing the identity on X0) is naturally identified with Hom(\Omega 1 X0=Y0 ; f \Lambda 0 I). Since \Omega 1 X0=Y0 is locally free of finite type, there is, for all i 2 Z, a canonical isomorphism (2.12.1) Ext i (\Omega 1 X0=Y0 ; f \Lambda 0 I) ' H i (X0; Hom(\Omega 1 X0=Y0 ; f \Lambda 0 I)) (and Hom(\Omega 1 X0=Y0 ; f \Lambda 0 I) ' TX 0=Y0 \Omega f \Lambda 0 I). If X0 is affine, the second term of (2.12.1) is zero for i * 1, and consequently there exists a lifting of X0 over Y , and two such liftings are isomorphic. 2.13. Here is an outline of the proof of 2.12. The data of a lifting X is equivalent to that of a cartesian square X0 j \Gamma ! X f0 # # f Y0 i \Gamma ! Y; with f smooth. let J be the ideal of thickness j. The flatness of f (2.10) implies that the homomorphism f \Lambda 0 I ! J induced from this square is an isomorphism. (It is moreover easy to verify that conversely, if X is a Y -extension of X0 by J such that the corresponding homomorphism f \Lambda 0 I ! J is an isomorphism, then X is automatically a lifting of X0.) Assertion (c) is therefore a particular case of 2.11 (b). The identification consists of associating with an automorphism u of X the "derivation" u\Gamma IdX . Similarly, if X1 and X2 are two liftings of X0, 2.11 (a) implies that X1 and X2 are isomorphisms if X0 is affine, and that the set of isomorphisms 6 In this section, when we speak of a lifting of a smooth Y0-scheme, it will be implicit, unless mentioned to the contrary, that we are thinking of it as a smooth lifting. 109 3. frobenius and cartier isomorphism of X1 over X2 is then an affine space under Hom(\Omega 1 X0=Y0 ; f \Lambda 0 I). Assertions (a) and (b) come about formally. The verification of (b) is analogous to that of 2.11: If X1 and X2 are two liftings of X0, the "difference" of their isomorphism classes is the class of the torsor under Hom(\Omega 1 X0=Y0 ; f \Lambda 0 I) of the local isomorphisms of X1 on X2. (We also observe that the classes of Y -extensions X1 and X2 of X0 by f \Lambda 0 I differ by a unique Y0-extension of X0 by f \Lambda 0 I, and invoke (2.5.1).) Finally, we indicate the construction of the obstruction !(f0), by assuming for simplicity that X0 is separated. First of all, by the jacobian criterion (2.8), the existence of a global lifting is assured in the case where X0 and Y0 are affine, and f0 is associated to a homomorphism of rings A0 ! B0, where B0 is the quotient of an A0-algebra of polynomials A0[t1; : : : ; tn] by the ideal generated by a sequence of elements (g1; : : : ; gr) such that the dgi are linearly independent at every point x of X0 (to arbitrarily lift the gi). Since (always according to (2.8)) f0 is locally of the preceding form, we can choose an open affine covering U = ((Ui)0)i2E of X0, and for each i, a lifting Ui of (Ui)0 over Y . Since X0 has been assumed separated, each intersection (Uij)0 = (Ui)0 " (Uj)0 is affine, and consequently, we can choose an isomorphism of liftings uij of Uij(U ij)0 over Ujj(U ij)0 . On a triple intersection (Uijk)0 = (Ui)0 " (Uj)0 " (Uk)0, the automorphism uijk = u \Gamma 1 ki ujkuij of Uij(U ijk)0 differs from the identity by a section cijk = u \Lambda ijk \Gamma Id of the sheaf Hom(\Omega 1 X0;Y0 ; f \Lambda 0 I). One verifies that (cijk) is a 2-cocycle of U with values in Hom(\Omega 1 X0;Y0 ; f \Lambda 0 I), where the class of this cocycle in H 2 (X0; Hom(\Omega 1 X0;Y0 ; f \Lambda 0 I)) does not depend on the choices, and that it vanishes if and only if on a refinement covering, the uij can be modified in a way in which they glue on the triple intersection, and also define a global lifting X of X0. This is the stated obstruction. Remark 2.14. The theory of gerbes [Gi] and that of the cotangent complex [I1], one or the other, allows us to get rid of the separation assumption made above, and especially gives a more conceptual proof of 2.12. 3. Frobenius and Cartier isomorphism The general references for this section are (SGA 5 XV 1) for the definitions and basic properties of Frobenius morphisms, absolute and relative, and [K1] 7 for the Cartier isomorphism (cf. also [I2] 0 2 and [D-I] 1). In this section, p denotes a fixed prime number. 3.1. We say that a scheme X is of characteristic p if pOX = 0, i.e. if the morphism X ! Spec Z factors (necessary in a unique way) through Spec Fp . If X is a scheme of characteristic p, we define the absolute Frobenius morphism of X (or, simply Frobenius endomorphism, if there is no fear of confusion) to be the endomorphism of X which is the identity over the underlying space of X, and the raising to the p-th power on OX. We denote it by FX . If X is affine with ring A; FX corresponds to the Frobenius endomorphism FA of A; a 7! a p . Let f : X ! Y be luc illusie, frobenius and hodge degeneration 110 a morphism of schemes. Then there is a commutative square (3.1.1) X FX \Gamma \Gamma ! X f # # f Y FY \Gamma \Gamma ! Y: Denote by X (p) (or X 0 , if there is no ambiguity) the scheme (Y; FY ) \Theta Y X induced from X by the change of base FY . The morphism FX defines a unique Y -morphism F = FX=Y : X ! X 0 , giving rise to a commutative diagram (3.1.2) X FX \Gamma \Gamma ! X 0 ! X f & # # f Y FY \Gamma \Gamma ! Y; where the upper composite is FX and the square is cartesian. We call F the relative Frobenius of X over Y . The morphisms of the upper line induce homeomorphisms on the underlying spaces (FY is a "universal homeomorphism", i.e. a homeomorphism and the remainder after any change of base). If Y is affine with ring A, and X is the affine space A n Y = Spec B, where B = A[t1; : : : ; tn], then X 0 = A n Y 7 , and the morphisms F : X ! X 0 and X 0 ! X correspond respectively to the homomorphisms ti 7! t p i and ati 7! a p ti (a 2 A). Proposition 3.2. Let Y be a scheme of characteristic p, and f : X ! Y a smooth morphism of pure relative dimension n (2.10). Then the relative Frobenius F : X ! X 0 is a finite and flat morphism, and the OX0-algebra F\Lambda OX is locally free of rank p n . In particular, if f is 'etale, F is an isomorphism, i.e. the square (3.1.1) is cartesian. We first treat the case where n = 0, which requires some commutative algebra: The point is that F is 'etale, because according to 2.6 (a), an 'etale Y -morphism between Y -schemes is automatically 'etale, and that a morphism which is both 'etale and radical 8 is an open immersion ((SGA 1 I 5.1) or (EGA IV 17.9.1)). Then the case where X is the affine space A n Y is immediate: The monomials Q t mi i , with 0 ^ mi ! p \Gamma 1 form a basis of F\Lambda OX over OX0. The general case is deduced from 2.7. Remarks 3.3. (a) Since, according to 2.10, \Omega i X=Y is locally free over OX of rank \Gamma n i \Delta , it follows from 3.2 that F\Lambda \Omega i X=Y is locally free over OX0 of rank p n \Gamma n i \Delta . (b) The statement of 3.2 relative to n = 0 admits a converse: If Y is of characteristic p and if X is a Y -scheme such that the relative Frobenius FX=Y is an isomorphism, then X is 'etale over Y (SGA 5 XV 1 Prop. 2). When Y is the spectrum of a field, this is "Mac Lane's criteria". 7 It is not true in general that X and X 0 are isomorphic as Y -schemes, it is the exceptional case here. 8 A morphism g : T ! S is said to be radical if g is injective and, for any point t of T , with image in S, the residue field extension k(s) ! k(t) is radical. 111 3. frobenius and cartier isomorphism 3.4. Let Y be a scheme of characteristic p and f : X ! Y a morphism. Set d = dX=Y (1.2.3). If s is a local section of OX , one has d(s p ) = ps p\Gamma 1 ds = 0. Since d(s p ) = F \Lambda X (ds) = F \Lambda (1 \Omega ds), it follows that (a) the canonical homomorphisms (1.3.2) associated to (FX ; FY ) and F , F \Lambda X \Omega 1 X=Y ! \Omega 1 X=Y ; F \Lambda \Omega 1 X0=Y ! \Omega 1 X=Y are zero; (b) the differential of the complex F\Lambda \Omega ffl X=Y is OX0-linear; in particular, the sheaves of cycles Z i , with boundaries B i and the cohomology H i = Z i =B i of the complex F\Lambda \Omega ffl X=Y are OX0 -modules, and the exterior product acting on the graded OX0- module L Z i F\Lambda \Omega ffl X=Y (resp. L H i F\Lambda \Omega ffl X=Y ) is a graded anti-commutative algebra. These facts are at the source of miracles of differential calculus in characteristic p. The principal result is the following theorem, due to Cartier [C] : Theorem 3.5. Let Y be a scheme of characteristic p and f : X ! Y a morphism. (a) There exists a unique homomorphism of graded OX -algebras fl : M \Omega i X0=Y ! M H i F\Lambda \Omega ffl X=Y ; satisfying the following two conditions : (i) for i = 0; fl is given by the homomorphism F \Lambda : OX0 ! F\Lambda OX; (ii) for i = 1; fl sends 1 \Omega ds to the class of s p\Gamma 1 ds in H 1 F\Lambda \Omega ffl X=Y (where 1 \Omega ds denotes the image of the section ds of \Omega 1 X=Y in \Omega 1 X0=Y . (b) If f is smooth, fl is an isomorphism. In case (b), fl is called the Cartier isomorphism, and is denoted by C \Gamma 1 . Its inverse, or the composite M Z i F\Lambda \Omega ffl X=Y ! M \Omega i X0=Y of its inverse with the projection of L Z i onto L H i , where Z i denotes the sheaf of cycles of F\Lambda \Omega ffl X=Y in degree i, is denoted by C. It is this latter homomorphism which was initially defined by Cartier, and which we sometimes call the Cartier operation. The adopted presentation in 3.5 is due to Grothendieck (handwritten notes), and detailed in [K1] 7. When Y is a perfect scheme, i.e. such that FY is an automorphism, for example if Y is the spectrum of a perfect field, one of the most significant cases for applications is this: If f is smooth, C \Gamma 1 gives by composition with the isomorphism M \Omega i X=Y ! M (FY )X \Lambda \Omega i X0=Y (where (FY )X ; X 0 ! X is the isomorphism induced from FY by change of base) an isomorphism C \Gamma 1 abs : M \Omega i X=Y ! M H i FX \Lambda \Omega ffl X=Y that we call the absolute Cartier isomorphism. Corollary 3.6. Let Y be a scheme of characteristic p and f : X ! Y a smooth morphism. Then for any i, the sheaves of OX0-modules F\Lambda \Omega i X=Y ; Z i F\Lambda \Omega ffl X=Y ; B i F\Lambda \Omega ffl X=Y ; H i F\Lambda \Omega ffl X=Y luc illusie, frobenius and hodge degeneration 112 are locally free of finite type (where Z i resp. B i denotes the sheaf of cycles resp. boundaries in degree i). Taking into account 3.3 (a) and the exactness of F\Lambda , it suffices to apply 3.5 (b), while proceeding by descending induction on i. We briefly indicate the proof of 3.5, according to [K1] 7. For (a), it amounts to the same, taking into account (1.3.2), to construct the composite of fl with the homomorphism L \Omega i X=Y ! L (FY )X \Lambda \Omega 1 X0=Y , i.e. a homomorphism of graded OX -algebras flabs : M \Omega i X=Y ! M H i FX \Lambda \Omega ffl X=Y satisfying the analogous conditions to (i) and (ii), i.e. given in degree zero by F \Lambda X , and in degree 1 sending ds to the class of s p\Gamma 1 ds. However the map of OX in H 1 FX \Lambda \Omega ffl X=Y sending a local section s of OX onto the class of s p\Gamma 1 ds is a Y -derivation (this is a result of the identity p \Gamma 1 ((X + Y ) p \Gamma X p \Gamma Y p ) = P 0^i^p p \Gamma 1 \Gamma p i \Delta X p\Gamma i Y i in Z[X; Y ]). By (1.2.4), it defines the desired homomorphism (flabs) 1 . Since the exterior algebra L \Omega i X=Y is strictly anti-commutative ("strictly" means to say that the elements of odd degree are of square zero), it is likewise of its sub-quotient L H i FX \Lambda \Omega ffl X=Y , and consequently there exists a unique homomorphism of graded algebras flabs extending the homomorphisms (flabs) 0 = F \Lambda X and (flabs) 1 . For (b), one can assume, according to 2.7, that f factors into X g \Gamma ! A n Y h \Gamma ! Y; where h is the canonical projection and g is 'etale. Given the square (3.1.1) relative to g, being cartesian according to 3.2, it is likewise the same of the analogous square with the relative Frobenius to Y (3.6.1) X F \Gamma ! X 0 g # # g 0 Z F \Gamma ! Z 0 ; where one sets for abbreviation A n Y = Z. According to 2.4 (b), the homomorphism g \Lambda \Omega i Z=Y ! \Omega i X=Y is an isomorphism. The square (3.6.1) being cartesian and F finite, thus furnishes an isomorphism of complexes of OX-modules (3.6.2) g 0\Lambda F\Lambda \Omega ffl Z=Y ! F\Lambda \Omega ffl X=Y : Since g 0 is 'etale, therefore flat, the homomorphism (3.6.3) g 0\Lambda H i F\Lambda \Omega ffl Z=Y ! H i F\Lambda \Omega ffl X=Y induced from (3.6.2) is an isomorphism. Since on the other hand g 0\Lambda \Omega i Z0=Y ! \Omega i X0=Y is an isomorphism (g 0 being 'etale), it follows (by functoriality of fl) that it suffices to prove (b) for Z. By analogous arguments (extension of scalars and K"unneth) one can easy reduce to Y = Spec Fp and n = 1, i.e. Z = Spec Fp [t]. Then Z 0 = Z, the monomials 1; t; : : : ; t p\Gamma 1 form a basis of F\Lambda OZ over OZ , and since the differential d : F\Lambda OZ ! F\Lambda \Omega 1 Z = (F\Lambda OZ)dt sends t i onto it i\Gamma 1 dt, one concludes that H 0 F\Lambda \Omega ffl Z=Fp (resp. H 1 F\Lambda \Omega ffl Z=Fp ) is free over OZ with basis 1 (resp. t p\Gamma 1 dt), and therefore that fl is an isomorphism. 113 3. frobenius and cartier isomorphism 3.7. There is a close link between Cartier isomorphism and Frobenius lifting. This was known by Cartier, and it serves as motivation for its construction. The decomposition and degeneration theorems of [D-I] originates from this, see n ffi 5. It consists of the following. Let i : T0 ! T be a thickening of order 1 and g0 : S0 ! T0 a flat morphism. By lifting to a T0-scheme S0 over T one extends a flat T -scheme over S equipped with a T0-isomorphism T0 \Theta T S ' S0, i.e. a cartesian square S0 j \Gamma ! S g0 # # g T0 i \Gamma ! T with g flat. If I (resp. J) is the ideal of thickening i (resp. j), the flatness of g implies that the canonical homomorphism g \Lambda 0 I ! J is an isomorphism (cf. 2.13). Take for i the thickening Spec Fp ! Spec Z=p 2 Z, of the ideal generated by p. Let Y0 be a scheme of characteristic p, and let Y be a lifting of Y0 over Z=p 2 Z. The ideal of Y0 in Y is therefore pOY , and the flatness of Y over Z=p 2 Z implies that multiplication by p induces an isomorphism (3.7.1) p : OY 0 , \Gamma ! pOY : Now let f0 : X0 ! Y0 be a smooth morphism of Fp -schemes. Denote by F0 : X0 ! X 0 0 the Frobenius of X0 relative to Y0. Assume given a (smooth) lifting X (resp. X 0 ) of X0 (resp. X 0 0 ) over Y and a Y -morphism F : X ! X 0 lifting F0, i.e. such that the square X0 ! X F0 # # F X 0 0 ! X 0 commutes. (We have seen that there exists obstructions to the existence of X; X 0 , and F , cf. 2.11 and 2.12, and that these objects, whenever they exist, are not unique. We will return to this later.) Proposition 3.8. Let f0 : X0 ! Y0 and F : X ! X 0 be given as in 3.7. Then: (a) multiplication by p induces an isomorphism p : \Omega 1 X0=Y0 , \Gamma ! p\Omega 1 X=Y : (b) the image of the canonical homomorphism F \Lambda : \Omega 1 X0=Y ! F\Lambda \Omega 1 X=Y is contained in pF\Lambda \Omega 1 X=Y . (c) Denote by 'F : \Omega 1 X p 0 =Y0 ! F0\Lambda \Omega 1 X0=Y0 luc illusie, frobenius and hodge degeneration 114 the homomorphism "induced from F \Lambda by division by p", i.e. the unique homomorphism rendering the square commutative \Omega 1 X0=Y F \Lambda \Gamma \Gamma ! pF0 \Lambda \Omega 1 X=Y # " p \Omega 1 X 0 0=Y0 ! F0 \Lambda \Omega 1 X0=Y0 : Then the image of 'F is contained in the kernel of the differential of the de Rham complex, i.e. in the sheaf of cycles Z 1 F0 \Lambda \Omega ffl X0=Y0 , and the composite of 'F with the projection on H 1 F0 \Lambda \Omega ffl X0=Y0 is the Cartier isomorphism C \Gamma 1 in degree 1 (cf. 3.5). Assertion (a) is trivial, (b) follows from 3.4 (a), and (c) is immediate from (a), (b) and the characterization of the Cartier isomorphism. Indeed, if a is a local section of OX , with reduction a0 module p, and a 0 lifts in OX0 the image a 0 0 of a0 in OX0 0 , we have F \Lambda a 0 = a p + pb for a local section b of OX . (This is because the reduction modulo p of F \Lambda a 0 is F \Lambda 0 a 0 0 = a p 0 .) Consequently F \Lambda (d 0 ) = pa p\Gamma 1 da + pdb; hence (3.8.1) 'F (da 0 0 ) = a p\Gamma 1 0 da0 + db0; for which (c) follows at once. 3.9. Suppose, as is in practice, and one of the more important cases, that Y0 is the spectrum of a perfect field k of characteristic p. Then the spectrum Y of the ring W2(k) of Witt vectors of length 2 over k lifts Y0 over Z=p 2 Z (besides, due to an isomorphism, it is the unique (flat) lifting of Y0). Recall that W2(k) is the set of pairs (a1; a2) of elements of k, equipped with addition and multiplication given by (a1; a2) + (b1; b2) = (a1 + b1; S2(a; b)); (a1; a2)(b1; b2) = (a1b1; P2(a; b)); where S2(a; b) = a2 + b2 + p \Gamma 1 (a p\Gamma 1 1 + b p\Gamma 1 1 \Gamma (a1 + b1) p ); P2(a; b) = b p 1 a2 + b2a p 1 : The homomorphism W2(k) ! k is given by (a1; a2) 7! a1. If k = Fp , then W2(k) ' Z=p 2 Z, the isomorphism being given by (a1; a2) 7! o/ (a1) + po/ (a2), where o/ denotes the multiplicative section of Z=p 2 Z ! Fp . (For an overall discussion of the theory of Witt vectors, see [S] II 6, [D-G] V.) In this case, if X0 is a smooth Y0-scheme (i.e. a smooth k-scheme), and since the absolute Frobenius of Y0 is an automorphism, lifting X0 over Y = Spec W2(k) is equivalent to lifting X 0 0 , and according to 2.12, the obstruction to the existence 115 4. derived categories and spectral sequences of such a lifting is found in Ext 2 (\Omega 1 X0 ; OX 0 ) ' H 2 (X0; TX 0 ) 9 . If this obstruction is zero, one can choose a lifting X 0 of X 0 0 and a lifting X of X0, and then the obstruction to a lifting F : X ! X 0 of the relative Frobenius F0 is found in Ext 1 (F \Lambda 0 \Omega 1 X 0 0 ; OX 0 ) ' Ext 1 (\Omega 1 X 0 0 ; F0 \Lambda OX 0 ) (2.11) 10 . In every case, these two obstructions are locally zero, and even as soon as X0 is affine. The choice of a lifting F furnishes then, according to 3.8, a relatively explicit description of the Cartier isomorphism in degree 1 (and therefore in every degree, by multiplicativity). 4. Derived categories and spectral sequences There are many reference sources on this subject at various levels. The reader with pressing obligations can consult [I3], which can be used as an introduction and contains a broad bibliography. We will limit ourselves here by recalling some fundamental points which we will use in the following section. 4.1. Let A be an abelian category (in practice, A will be the category of OXmodules of a scheme X). We denote by C(A) the category of A-complexes, with differential of degree 1, and further denote by L ffl (or L) for such a complex \Delta \Delta \Delta ! L i ! L i+1 ! \Delta \Delta \Delta : We say that L is with lower bounded degree (resp. upper, resp. with bounded degree) if L i = 0 for i sufficiently small (resp. sufficiently large, resp. outside of a bounded interval of Z). We denote by Z i L = Ker d : L i ! L i+1 ; B i L = Im d : L i\Gamma 1 ! L i ; H i L = Z i L=B i L, respectively the objects of cycles, boundaries and cohomology in degree i. If A is the category of OX -modules, we write C(X) in place of C(A), and often H i L instead of H i L for an object of C(X) (in order to indicate that it acts on the cohomology sheaf in degree i, and not on the global cohomology group H i (X; L)). For n 2 Z, the naive truncation L ^n (resp. L *n ) of a complex L is the quotient (resp. the subcomplex) of L which coincides with L in degree ^ n (resp. * n) and has zero components elsewhere. The canonical truncation o/^nL (resp. o/*nL) is the subcomplex (resp. quotient) of L with components L i for i ! n; Z i L for i = n and 0 for i ? n (resp: L i for i ? n; L i =B i L for i = n and 0 for i ! n). One sets o/!nL = o/^n\Gamma 1L. The inclusion o/^nL ,! L induces an isomorphism on H i for i ^ n. The projection L i o/*nL induces an isomorphism on H i for i * n. For n 2 Z, the translate L[n] of a complex L is the complex with components L[n] i = L n+i and with differential dL[n] = (\Gamma 1) n dL. A complex L is said to be concentrated in degree r (resp. in the interval [a; b]) if L i = 0 for i 6= r (resp. i =2 [a; b]). An object E of A is often considered as a complex concentrated in degree zero. The complex E[\Gamma n] is then concentrated in degree n, with component E in this degree. 9 We omit here, for abbreviation, =Y0 in the notation of differentials. 10 One can show ( [Me-Sr] Appendix) that the obstruction to a choice of (X; X 0 ; F ) such that X 0 is the inverse image of X by the Frobenius automorphism of W2(k) is found in Ext 1 (\Omega X0 0 ; B 1 F\Lambda \Omega ffl X0 ); more precisely, such a triplet (X; X 0 ; F ) exists if and only if the extension class 0 ! B 1 F\Lambda \Omega ffl X0 ! Z 1 F\Lambda \Omega ffl X0 C \Gamma ! \Omega 1 X0 0 ! 0 (particular case i = 1 of the Cartier isomorphism 3.5) is zero. See [Sr] for an application to another proof of the principal theorem of [D-I]. luc illusie, frobenius and hodge degeneration 116 A homomorphism of complexes u : L ! M is called a quasi-isomorphism if H i u is an isomorphism for all i. We say that a complex K is acyclic if H i K = 0 for all i. If u : L ! M is a homomorphism of complexes, the cone N = C(u) of u is the complex defined by N i = L i+1 \Phi M i , with differential d(x; y) = (\Gamma dLx; ux + dM y). For that u to be a quasi-isomorphism, it is necessary and sufficient that C(u) is acyclic. 4.2. Denote by K(A) the category of complexes of A up to homotopy, i.e. the category having the same objects as C(A) but for which the set of arrows of L in M is the set of homotopy classes of morphisms of L into M . The derived category of A, denoted by D(A), is the category induced from K(A) by formally reversing the (homotopy classes of) quasi-isomorphisms: The quasi-isomorphisms of K(A) become isomorphisms in D(A) and D(A) is universal for this property. When A is the category of OX -modules over a ringed space X, we write D(X) instead of D(A). The categories K(A) and D(A) are additive categories, and one has canonical additive functors C(A) ! K(A) ! D(A): The category D(A) has the same objects as C(A). Its arrows are calculated "by fractions" from those of K(A): An arrow u : L ! M of D(A) is defined by a couple of arrows of C(A) of the type L s \Gamma L 0 f \Gamma ! M or L g \Gamma ! M 0 t \Gamma M; where s and t are quasi-isomorphisms. More precisely, one shows that the homotopy classes of quasi-isomorphisms with source M (resp. target L) form a filtered category 11 (resp. the opposite of a filtered category) and that one has HomD(A)(L; M ) = lim \Gamma ! t:M!M0 HomK(A)(L; M 0 ) = lim \Gamma ! s:L0!L HomK(A)(L 0 ; M ) as t (resp. s) runs over the preceeding category (resp. its opposite). If L; M are complexes, we set, for i 2 Z, Ext i (L; M ) = HomD(A)(L; M [i]) = HomD(A)(L[\Gamma i]; M ): The functors H i and the canonical truncation functors o/^i; o/*i on C(A) naturally extend to D(A). On the other hand, it is not the same as the naive truncation functors. 4.3. We denote by D + (A) (resp. D \Gamma (A), resp. D b (A)) the full subcategory of D(A) formed from complexes L cohomologically bounded below (resp. above, resp. bounded), i.e. such that H i L = 0 for i small enough (resp. large enough, resp. outside a bounded interval). If A contains sufficiently many injectives (i.e. if any object of A embeds in an injective), for example if A is the category of OXmodules over a scheme X, then any object of D + (A) is isomorphic to a complex, with bounded below degree, formed from injectives, and the category D + (A) is equivalent to the full subcategory of K(A) formed from such complexes. 11 A category I is said to be filtered if it satisfies the following conditions (a) and (b): (a) For any two arrows f; g : i ! j, there exists an arrow h : j ! k such that hf = hg. (b) Assume given any objects i and j, there exists an object k and arrows f : i ! k, g : j ! k. 117 4. derived categories and spectral sequences 4.4. The categories K(A) and D(A) are not in general abelian, but possess a triangle category structure, in the sense of Verdier [V]. This structure is defined by the family of distinguished triangles. A triangle is a sequence of arrows T = (L ! M ! N ! L[1]) of K(A) (resp. D(A)). A morphism of T in T 0 = (L 0 ! M 0 ! N 0 ! L 0 [1]) is a triplet (u : L ! L 0 ; v : M ! M 0 ; w : N ! N 0 ) such that the three squares formed with u; v; w; u[1] commute. A triangle is said to be distinguished if it is isomorphic to a triangle of the form L u \Gamma ! M i \Gamma ! C(u) p \Gamma ! L[1]; where u is the cone of a morphism of complexes u, and i (resp. p) denotes the obvious inclusion (resp. the opposite of the projection). Any short exact sequence of complexes 0 ! E u \Gamma ! F ! G ! 0 defines a distinguished triangle D(A), by means of the natural quasi-isomorphism C(u) ! G, and any distinguished triangle of D(A) is isomorphic to a triangle of this type. Any distinguished triangle T = (L ! M ! N ! L[1]) of D(A) gives rise to a long exact sequence \Delta \Delta \Delta ! H i L ! H i M ! H i N d \Gamma ! H i+1 L ! \Delta \Delta \Delta ; \Delta \Delta \Delta ! Ext i (E; L) ! Ext i (E; M ) ! Ext i (E; N ) ! Ext i+1 (E; L) ! \Delta \Delta \Delta ; \Delta \Delta \Delta ! Ext i (N; E) ! Ext i (M; E) ! Ext i (L; E) ! Ext i+1 (N; E) ! \Delta \Delta \Delta ; for E 2 ob D(A). If the triangle T is associated to a short exact sequence given explicitly above, the operator d of the first of these sequences is the usual boundary operator (this is the reason for the convention of sign in the definition of p). 4.5. Let L be a complex of A and i 2 Z. The quotient o/^iL=o/^i\Gamma 1L is mapped quasi-isomorphically onto H i L[\Gamma i]. Therefore there is a canonical distinguished triangle from D(A) o/^i\Gamma 1L ! o/^iL ! H i L[\Gamma i] ! o/^i\Gamma 1L[1]: We similarly define a canonical distinguished triangle H i\Gamma 1 L[\Gamma i + 1] ! o/*i\Gamma 1L ! o/*iL ! H i\Gamma 1 L[\Gamma i + 2]: Finally, o/[i\Gamma 1;i]L := o/*i\Gamma 1o/^iL = o/^io/*i\Gamma 1L = (0 ! L i\Gamma 1 =B i L ! Z i L ! 0) defines a distinguished triangle H i\Gamma 1 L[\Gamma i + 1] ! o/[i\Gamma 1;i]L ! H i L[\Gamma i] !; which furnishes a canonical element (4.5.1) ci 2 Ext 2 (H i L; H i\Gamma 1 L): The triplet (H i\Gamma 1 L; H i L; ci) is an invariant of L in D(A). It permits its reconstruction up to an isomorphism if L is cohomologically concentrated in degree i \Gamma 1 and i. One can show that the ci universally realizes the differential d2 of the spectral sequences of derived functors applied to L (cf. Verdier's theorem 12 , or [D3]). 12 Which should be appearing soon in Ast'erique. luc illusie, frobenius and hodge degeneration 118 4.6. Let L be an object of D b (A). We say that L is decomposable if L is isomorphic, in D(A), to a complex with zero differential. If L is decomposable, and if u : L 0 ! L is an isomorphism of D(A), with L 0 having zero differential, then u induces isomorphisms L 0i ! H i L. In particular L 0 has bounded degree and L 0 = L L 0i [\Gamma i] (in C(A)) (4.1), therefore (4.6.1) L ' M H i L[\Gamma i] (in D(A)). Conversely, if L satisfies (4.6.1), L is trivially decomposable. If L is decomposable, one calls a decomposition of L the choice of an isomorphism (4.6.1) inducing the identity on H i for all i. There exists a finite sequence of obstructions to the decomposability of L: The first are the classes ci (4.5.1); if the ci are zero, there are secondary obstructions in Ext 3 (H i L; H i\Gamma 2 L), etc. In addition, if L is decomposable, L admits in general many decompositions. In the following section, we are especially interested in the case when L is concentrated in degree 0 and 1 : L = (L 0 ! L 1 ). In this case: a) the class c1 2 Ext 2 (H 1 L; H 0 L) is the obstruction to the decomposability of L; b) the giving of a decomposition of L is equivalent to that of a morphism H 1 L[\Gamma 1] ! L inducing the identity on H 1 ; c) The set of decompositions of L is an affine space under Ext 1 (H 1 L; H 0 L) ([D-I] 3.1). 4.7. We now return for example to [H1], II for the definition of the derived functors L \Omega ; RHom 13 , R Hom; Rf\Lambda ; Lf \Lambda ; R\Gamma in the derived category D(X), where X is a variable scheme, and the description of certain remarkable relations between these functors. We need only recall that these functors are, compared to each argument, exact functors, i.e. transform distinguished triangles to distinguished triangles, and are "calculated" in the following way: (a) For E 2 ob D(X); F 2 ob D \Gamma (X); E L \Omega F ' E \Omega F 0 if F ' F 0 in D(X), with F 0 having upper bounded degree (4.1) and with flat components. For given F , there exists a quasi-isomorphism F 0 ! F with F 0 of the preceding type; moreover the homotopy classes of such quasi-isomorphisms form a coinitial system (in the category of classes of quasi-isomorphisms with target F , cf. 4.2). (b) For E 2 ob D(X); F 2 ob D + (X), if F ' F 0 , with F 0 having lower bounded degree and with injective components, then RHom(E; F ) ' Hom ffl (E; F 0 ) and R Hom(E; F ) ' Hom ffl (E; F 0 ). For given F , there exists a quasi-isomorphism F ! F 0 with F 0 of the preceding type (and the homotopy classes of such quasi-isomorphisms form a cofinal system). (c) For f : X ! Y and E 2 ob D + (X), if E ' E 0 , with E 0 having lower bounded degree and with flasque components (for example, injective), then Rf\Lambda E ' f\Lambda E 0 and R\Gamma (X; E) ' \Gamma (X; E 0 ). One simply writes H i (X; E) instead of H i R\Gamma (X; E); and more generally, one defines in the same way, Rf\Lambda : D + (X; f \Gamma 1 (OY )) ! D + (Y ), where D(X; f \Gamma 1 (OY )) denotes the derived category of the category of complexes of f \Gamma 1 (OY )-modules (the de Rham complex \Omega ffl X=Y is such a complex). (d) For f : X ! Y and F 2 ob D \Gamma (Y ); Lf \Lambda F ' f \Lambda F 0 if F ' F 0 , with F 0 having upper bounded degrees and with flat components. 13 An error of sign slipped into the definition of the complex Hom ffl (L; M) in [H1] p. 64: For u 2 Hom(L i ; M i+n ), it necessarily reads du = d ffi u + (\Gamma 1) n+1 u ffi d. 119 5. decomposition, degeneration and vanishing theorems 4.8. It can be said that spectral sequences are perhaps one of the most avoided objects in mathematics, and yet at the same time, are one of the most useful algebraic tools for cohomology. This is particularly true of derived categories, which sometimes contributes to this, but they remain essential. There are many references, the oldest ([C-E], XV) being one of the best. In these notes, we will be especially interested in the spectral sequence called the Hodge to de Rham, for which we will recall the definition. Let T : A ! B be an additive functor between abelian categories. Assume that A has sufficiently many injectives. Then T admits a right derived functor RT : D + (A) ! D + (B); which is calculated by RT (K) ' T (K 0 ) if K ! K 0 is a quasi-isomorphism with K 0 with bounded below degree and with injective components. The objects of cohomology H i ffi RT : D + (A) ! B are denoted by R i T . For K 2 ob D(A), with bounded below degree, there is a spectral sequence (4.8.1) E ij 1 = R j T (K i ) ) R \Lambda T (K); called the first spectral sequence of hypercohomology of T . It is obtained in the following way: Chooses a resolution K ! L of K by a bicomplex L, such that each column L iffl is an injective resolution of K i . If sL denotes the associated simple complex, the resulting homomorphism of complexes K ! sL is a quasiisomorphism, therefore RT (K) ' T (sL) = sT (L); RT (K i ) ' T (L iffl ), and the filtration of sT (L) by the first degree of L given rise to (4.8.1). Let K be a field and X a k-scheme. The group (cf. (1.7.1) and 4.7 (c)) (4.8.2) H i DR (X=k) = H i (X; \Omega ffl X=k ) = \Gamma (Spec k; R i f\Lambda (\Omega ffl X=k )) (where f : X ! Spec k is the structure morphism) is called i-th de Rham cohomology group of X=k. This is a k-vector space. The spectral sequence (4.8.1) relative to the functor \Gamma (X; ffl) and the complex \Omega ffl X=k is called the Hodge to de Rham spectral sequence of X=k : (4.8.3) E ij 1 = H j (X; \Omega i X=k ) ) H \Lambda DR (X=k): This is a spectral sequence of k-vector spaces. The groups H j (X; \Omega i X=k ) are called the Hodge cohomology groups of X over k. If X is proper over k ([H2] II 4) (for example, projective over k, i.e. a closed subscheme of a projective space P n k ), and since the \Omega i X=k are coherent sheaves (2.1), the finiteness theorem of SerreGrothendieck ([H2] III 5.2 in the projective case, (EGA III 3) in the general case) implies that the Hodge cohomology groups of X over k are finite dimensional kvector spaces. By the spectral sequence (4.8.3), it follows from this that the de Rham cohomology groups H n DR (X=k) are also finite dimensional over k. Moreover, for each n, one has (4.8.4) X i+j=n dimk H j (X; \Omega i X=k ) * dimk H n DR (X=k); with equality for all n if and only if the Hodge to de Rham spectral sequence of X over k degenerates at E1, i.e. the differential dr is zero for all r * 1. luc illusie, frobenius and hodge degeneration 120 5. Decomposition, degeneration and vanishing theorems in characteristic p ? 0p ? 0p ? 0 In this section, as in n ffi 3; p will denote a fixed prime number. The main result is the following theorem ([D-I] 2.1, 3.7): Theorem 5.1. Let S be a scheme of characteristic p. Assume given a (flat) lifting T of S over Z=p 2 Z (3.7). Let X be a smooth S-scheme, and let us denote as in 3.1, F : X ! X 0 the relative Frobenius of X=S. Then if X 0 admits a (smooth) lifting over T , the complex of OX0 -modules o/!pF\Lambda \Omega ffl X=S (4.1) is decomposable in the derived category D(X 0 ) of OX0-modules (4.6). 5.2. Before beginning the proof, note that a decomposition of o/!pF\Lambda \Omega ffl X=S is equivalent to giving an arrow of D(X 0 ) M i!p H i F\Lambda \Omega ffl X=S [\Gamma i] ! F\Lambda \Omega ffl X=S inducing the identity on H i for all i ! p. According to Cartier's theorem (3.5), this data is still equivalent to that of an arrow of D(X 0 ) (5.2.1) ' : M i!p \Omega i X0=S [\Gamma i] ! F\Lambda \Omega ffl X=S inducing C \Gamma 1 on H i for all i ! p. The proof in fact consists of associating canonically such an arrow ' to each lifting of X 0 over T . It includes three steps. Step A. We start by treating the case where F admits a global lifting. Proposition 5.3. Under the hypothesis of 5.1, assume that F : X ! X 0 admits a global lifting G : Z ! Z 0 , where Z (resp. Z 0 ) lifts X (resp. X 0 ) over T . Let (5.3.1) 'G : M \Omega i X0=S [\Gamma i] ! F\Lambda \Omega ffl X=S be the homomorphism of complexes, with i-th component ' i G , defined in the following way: ' 0 G = F \Lambda : OX ! F\Lambda OX ; ' 1 G : \Omega 1 X0=S ! F\Lambda \Omega 1 X=S is the homomorphism "G \Lambda =p" defined in 3.8 (c). For i * 1; ' i G is composed with \Lambda i ' 1 G and of the product \Lambda i F\Lambda \Omega 1 X=S ! F\Lambda \Omega i X=S . Then 'G is a quasi-isomorphism, inducing the Cartier isomorphism C \Gamma 1 on H i for all i. This is immediate. Step B. This is the principal step. We show that the giving of a lifting Z 0 of X 0 over T allows us to define a decomposition of o/^1F\Lambda \Omega ffl X=S , i.e. a homomorphism ' 1 Z0 : \Omega 1 X0=S [\Gamma 1] ! F\Lambda \Omega ffl X=S of D(X 0 ) (and not C(X 0 )) inducing C \Gamma 1 over H 1 . With this intention, we need to compare the homomorphisms ' 1 G of (5.3.1) associated to any other lifting of F with target Z 0 . 121 5. decomposition, degeneration and vanishing theorems Lemma 5.4. To any pair (G1 : Z1 ! Z 0 ; G2 : Z2 ! Z 0 ) of liftings of F is associated canonically a homomorphism (5.4.1) h(G1; G2) : \Omega 1 X0=S ! F\Lambda OX such that ' 1 G2 \Gamma ' 1 G1 = dh(G1; G2). If G3 : Z3 ! Z 0 is a third lifting of F , one has (5.4.2) h(G1; G2) + h(G2; G3) = h(G1; G3): Let us suppose initially that Z1 and Z2 are isomorphic (in the sense of 2.12 (b)). Choose an isomorphism u : Z1 , \Gamma ! Z2. Then G2u and G1 lift F , i.e. extend to Z1 the composite X F \Gamma ! Z ,! Z 0 . Therefore according to 2.11 (b), they differ by a homomorphism hu of F \Lambda \Omega 1 X0=S in OX , or what amounts to the same, of \Omega 1 X0=S in F\Lambda OX . If v is a second isomorphism of Z1 onto Z2, then taking into account 3.4 (a), it follows from 2.11 (b) that u and v differ by a homomorphism "u \Gamma v" : \Omega 1 X=S ! OX, therefore G2u and G2v differ by the composite of "u \Gamma v" and the homomorphism F \Lambda \Omega 1 X0=S ! \Omega 1 X=S , which is zero, a fortiori G2u = G2v. Therefore hu does not depend on the choice of u. Since Z1 and Z2 are locally isomorphic according to 2.11 (a), we deduce from this a homomorphism (5.4.1) characterized by the property that if u is an isomorphism of Z1 onto Z2 over an open subset U of X (recall still that Z1; Z2 and X have the same underlying space), the restriction of h(G1; G2) to U is the homomorphism hu, the "difference" between G1 and G2u. The formula ' 1 G2 \Gamma ' 1 G1 = dh(G1; G2) follows from the explicit description of ' 1 G given in (3.8.1), and formula (5.4.2) is immediate. Now fix the lifting Z 0 of X 0 over T . According to 2.11 (a) and 2.12 (a), we can choose an open covering U = (Ui)i2I of X in such a way that we have for each i, a lifting Zi of Ui over T and a lifting Gi : Zi ! Z 0 of FjU i . We then arrange for each i, a homomorphism of complexes fi = ' 1 Gi : \Omega 1 X0=SjUi [\Gamma 1] ! F\Lambda \Omega ffl X=SjUi 14 of (5.3.1), and for each pair (i; j), a homomorphism hij = h(GijU ij ; GjjU ij ) : \Omega 1 X0=SjUij ! F\Lambda \Omega ffl X=SjUi of (5.4.1), where Uij = Ui " Uj. These datum are connected by fj \Gamma fi = dhij (on Uij); hij + hjk = hik (on Uijk = Ui " Uj " Uk): They make it possible to define a homomorphism of complexes of OX0 -modules ' 1 Z0;(U;(Gi)) : \Omega 1 X0=S [\Gamma 1] ! ^ C(U; F\Lambda \Omega ffl X=S ); where ^ C(U; F\Lambda \Omega ffl X=S ) is the simple complex associated to the ^ Cech bicomplex of the covering U with values in F\Lambda \Omega ffl X=S . The components of this complex are given by ^ C(U; F\Lambda \Omega ffl X=S ) n = M a+b=n ^ C b (U; F\Lambda \Omega a X=S ) 14 We identify the underlying spaces of X and X 0 by means of F (3.1). luc illusie, frobenius and hodge degeneration 122 with differential d = d1 + d2, where d1 is induced by the differential of the de Rham complex and d2 is, in bidegree (a; b), equal to (\Gamma 1) a ( P (\Gamma 1) i @ i ) (see [G] 5.2 or [H2] III 4.2). In particular, ^ C(U; F\Lambda \Omega ffl X=S ) 1 = ^ C 1 (U; F\Lambda OX ) \Phi ^ C 0 (U; F\Lambda \Omega 1 X=S ): The morphism ' 1 Z0;(U;(Gi)) is defined as having for components, ('1; '2) in degree 1, with ('1!)(i; j) = hij(!)jU ij ; ('2!)(i) = fi(!)jU i : Using the fact that the fi are morphisms of complexes, together with the above formulas connecting the fi and the hij, it follows that ' 1 Z0;(U;(Gi)) is thus a well-defined morphism of complexes. We also has at our disposal the natural augmentation ffl : F\Lambda \Omega ffl X=S ! ^ C(U; F\Lambda \Omega ffl X=S ); which is a quasi-isomorphism, because for any a, the complex C(U; F\Lambda \Omega a X=S ) is a resolution of F\Lambda \Omega a X=S (cf. [Go] or [H2] loc. cit.). We then define ' 1 Z0 : \Omega 1 X0=S [\Gamma 1] ! F\Lambda \Omega ffl X=S to be the arrow of D(X 0 ) composed with ' 1 Z0;(U;(Gi)) and with the inverse of ffl (4.2). If (U = (Ui)i2I ; (Gi)i2I ) and (V = (Vj )j2J ; (Gj )j2J ) are two choices of systems of Frobenius liftings, then by considering the covering U ` V, indexed by I ` J, formed from the Ui and from Vj, it follows that ' 1 Z0 does not depend on choices (cf. [D-I] p. 253). Moreover ' 1 Z0 induces C \Gamma 1 on H 1 : The question is indeed local, therefore we can arrange for a global lifting of F , and apply 5.3. This completes step B. Step C. We again fix a lifting Z 0 of X 0 , and show how to extend the decomposition of o/^1F\Lambda \Omega ffl X=S defined by ' i Z0 (i = 0; 1) to a decomposition of o/!pF\Lambda \Omega ffl X=S . We use for this the multiplicative structure of the de Rham complex. From ' 1 Z0 we deduce, for all i * 1, an arrow of D(X 0 ) (' 1 Z0 ) L \Omega i = ' 1 Z0 L \Omega \Delta \Delta \Delta L \Omega ' 1 Z0 : (\Omega 1 X0=S [\Gamma 1]) L \Omega i ! (F\Lambda \Omega ffl X=S ) L \Omega i : Since \Omega 1 X0=S is locally free of finite type, we have (4.7 (a)) (\Lambda ) (\Omega 1 X0=S [\Gamma 1]) L \Omega i ' (\Omega 1 X0=S ) \Omega i [\Gamma i]; and similarly, since the F\Lambda \Omega a X=S are locally free of finite type (3.3 (a)), (\Lambda \Lambda ) (F\Lambda \Omega ffl X=S ) L \Omega i ' (F\Lambda \Omega ffl X=S ) \Omega i : We then define for i ! p, ' i Z0 : \Omega i X0=S [\Gamma i] ! F\Lambda \Omega ffl X=S as the composite (via (\Lambda ) and (\Lambda \Lambda )) of the standard antisymmetrization arrow \Omega i X0=S [i] ! (\Omega 1 X0=S ) \Omega i [\Gamma i]; !1 ^ \Delta \Delta \Delta ^ !i 7! 1 i! X oe2Gi sgn(oe)!oe(1) ^ \Delta \Delta \Delta ^ !oe(i) 123 5. decomposition, degeneration and vanishing theorems (well defined because of the assumption i ! p), of the arrow (' 1 Z0 ) L \Omega i , and of the product arrow (F\Lambda \Omega ffl X=S ) \Omega i ! F\Lambda \Omega ffl X=S . Since the antisymmetrication arrow is a section of the projection of (\Omega 1 X0=S ) \Omega i onto \Omega i X0=S , the multiplicative property of the Cartier isomorphism results in ' i Z0 inducing C \Gamma 1 over H i , and this completes the proof of the theorem. Taking into account 3.9, we then deduce: Corollary 5.5. Let k be a perfect field of characteristic p, and let X be a smooth scheme over S = Spec k. If X is lifted over T = Spec W2(k), then o/!pF\Lambda \Omega ffl X=S is decomposable in D(X 0 ). Moreover, if X is of dimension ! p, then F\Lambda \Omega ffl X=S is decomposable. Remark 5.5.1. According to 5.3, if X is smooth over Spec k and if X and F are lifted over W2(k), then F\Lambda \Omega ffl X=S is decomposable (and this is without the assumption of dimension on X). This is the case for example if X is affine. On the other hand, if X is proper, it is rare that X admits a lifting over W2(k) where F is lifted. One can show that if X and F are lifted, then X is ordinary, i.e. satisfies H j (X; B i \Omega ffl X=S ) = 0 for all (i; j) (cf. 8.6). The notion of an ordinary variety, which makes sense only in non-zero characteristic, was initially introduced for curves and abelian varieties. It intervenes in rather many questions in algebraic geometry. See [I4] for an introduction and the references cited there. Corollary 5.6. Let k be a perfect field of characteristic p, and let X be a smooth and proper k-scheme, of dimension ! p. If X is lifted over W2(k), the spectral sequence of Hodge to de Rham (4.8.3) of X over k E ij 1 = H j (X; \Omega i X=k ) ) H \Lambda DR (X=k) degenerates at E1. By virtue of the compatibility of \Omega i by a change of base (1.3.2), the absolute Frobenius isomorphism FS : S ! S (where S = Spec k) induces, for all (i; j), an isomorphism F \Lambda S H j (X; \Omega i X=k ) , \Gamma ! H j (X 0 ; \Omega i X0=k ), and in particular, we have dimk H j (X; \Omega i X=K ) = dimk H j (X 0 ; \Omega i X0=k ): In addition, since F : X ! X 0 is a homeomorphism, one has canonically, for all n, H n (X 0 ; F\Lambda \Omega ffl X=k ) , \Gamma ! H n (X; \Omega ffl X=k ) = H n DR (X=k): Finally, if X is lifted over W2(k), a decomposition ' : L \Omega i X0=S [\Gamma i] , \Gamma ! F\Lambda \Omega ffl X=S of F\Lambda \Omega ffl X=S in D(X 0 ) induces, for all n, an isomorphism M i+j=n H j (X 0 ; \Omega i X0=k ) , \Gamma ! H n (X 0 ; F\Lambda \Omega ffl X=k ): It follows from this that one has, for all n, X i+j=n dimk H j (X; \Omega i X=k ) = dimk H n DR (X=k); and according to 4.8, this results in the degeneration at E1 of the Hodge to de Rham spectral sequence. luc illusie, frobenius and hodge degeneration 124 5.7. For the remaining part to follow, the reader can consult [H2] II, III. Let k be a ring and X a projective k-scheme, i.e. admits a closed k-immersion i in a standard projective space P = P r k = Proj k [t0; : : : ; tn]. Let L be an invertible sheaf over X. Recall that: (i) L is very ample if one has L ' i \Lambda OP (1) for such a closed immersion i, which means that there exists global sections sj 2 \Gamma (X; L) (0 ^ j ^ r) defining a closed immersion x 7! (s0(x); : : : ; sr(x)) of X in P ; (ii) L is ample, if there exists n ? 0 such that L \Omega n is very ample. Assume L ample. Then, according to Serre's theorem ([H2] II 5.17, III 5.2): (a) For any coherent sheaf E on X, there exists an integer n0 such that for any n * n0; E \Omega L \Omega n is generated by a finite number of its global sections, i.e. a quotient of O N X for suitable N . (b) For any coherent sheaf E on X, there exists an integer n0 such that for any n * n0 and all i * 1, one has H i (X; E \Omega L \Omega n ) = 0: The theorem which follows is an analog in characteristic p, of the Kodaira-AkizukiNakano vanishing theorem [KAN], [AkN]: Theorem 5.8. Let k be a field of characteristic p, and let X be a smooth projective k-scheme. Let L be an ample invertible sheaf on X. Then if X is of pure dimension d ! p (cf. 2.10) and is lifted over W2(k), we have H j (X; L \Omega \Omega i X=k ) = 0 for i + j ? d;(5.8.1) H j (X; L \Omega \Gamma 1 \Omega \Omega i X=k ) = 0 for i + j ! d:(5.8.2) This is a corollary of 5.5, due to Raynaud. The proof is analogous to that of 5.6, starting from 5.5. First of all, by the Serre duality theorem ([H2] III 7.7, 7.12), if M is an invertible sheaf on X, and if i+i 0 = d = j +j 0 , then the finite dimensional k-vector spaces H j (X; M \Omega \Omega i X=k ) and H j 0 (X; M \Omega \Gamma 1 \Omega \Omega i 0 X=k ) are canonically dual. Formulas (5.8.1) and (5.8.2) are therefore equivalent. It will be more convenient to prove (5.8.2). By Serre's vanishing theorem (5.7 (b)), there exists n * 0 such that H j (X; L \Omega p n \Omega \Omega i X=k ) = 0 for all j ? 0 and all i. By Serre duality, it follows that H j (X; L \Omega \Gamma p n \Omega \Omega i X=k ) = 0 for all j ! d and all i, and in particular for all (i; j) such that i + j ! d. Proceeding by descending induction on n, it therefore suffices to prove the following assertion: (\Lambda ) if M is an invertible sheaf over X satisfying H j (X; M \Omega p \Omega \Omega i X=k ) = 0 for all (i; j) such that i + j ! d, then H j (X; M \Omega \Omega i X=k ) = 0 for all (i; j) such that i + j ! d. Note as in 5.1, X 0 is the scheme induced from X by the change of base by the absolute Frobenius of S = Spec k. If FX denotes the absolute Frobenius of X, we have a canonical isomorphism F \Lambda X M ' M \Omega p , induced by the map m 7! m \Omega p , and therefore an isomorphism F 0 \Lambda M 0 ' M \Omega p , where F : X ! X 0 is the relative Frobenius and M 0 is the inverse image of M over X 0 . We deduce, for all i, the following isomorphisms of OX0-modules (\Lambda \Lambda ) M 0 \Omega F\Lambda \Omega i X=k ' F\Lambda (F \Lambda M 0 \Omega \Omega 1 X=k ) ' F\Lambda (M \Omega p \Omega \Omega i X=k ): 125 5. decomposition, degeneration and vanishing theorems Let us consider the spectral sequence (4.8.1) relative to the functor T = \Gamma (X 0 ; ffl) and on the complex K = M 0 \Omega F\Lambda \Omega ffl X=k : E ij 1 = H j (X 0 ; M 0 \Omega F\Lambda \Omega i X=k ) ) H \Lambda (X 0 ; M 0 \Omega F\Lambda \Omega ffl X=k ): The hypothesis and (\Lambda \Lambda ) imply that E ij 1 = 0 for i + j ! d. Therefore H n (X 0 ; M 0 \Omega F\Lambda \Omega ffl X=k ) = 0 for n ! d: But like, according to 5.5, F\Lambda \Omega ffl X=k is decomposable, we have (in D(X 0 )) F\Lambda \Omega ffl X=k ' M \Omega i X0=k [\Gamma i]; therefore H n (X 0 ; M 0 \Omega F\Lambda \Omega ffl X=k ) ' M i+j=n H j (X 0 ; M 0 \Omega \Omega i X0=k ); and therefore H j (X 0 ; M 0 \Omega \Omega i X0=k ) = 0 for i + j ! d: The conclusion (\Lambda ) follows from this, since we have F \Lambda S H j (X; M \Omega \Omega i X=k ) ' H j (X 0 ; M 0 \Omega \Omega i X0=k ) (cf. the end of the proof of 5.6). Remarks 5.9. The reader will find in [D-I] many complements of the aforementioned results. Here are some. 1. Let us assume given the hypothesis of 5.1. Then: (a) X 0 is lifted over T if and only if o/^1F\Lambda \Omega ffl X=S is decomposable in D(X 0 ) (or, what amounts to the same, o/!pF\Lambda \Omega ffl X=S is). Recall that there exists an obstruction ! 2 Ext 2 (\Omega 1 X0=S ; OX0) to the lifting of X 0 (2.12 (a) and 3.7.1)), and that taking into account the Cartier isomorphism, this is in the same group that is found the obstruction c1 to the decomposability of o/^1F\Lambda \Omega ffl X=S (4.6(a)): One can show with some convenient conventions of signs, that ! = c1. (b) If X 0 is lifted over T , the set of isomorphism classes of liftings of X 0 is an affine space under Ext 1 (\Omega 1 X0=S ; OX) (2.12 (b) and (3.7.1)), and (always taking into account the Cartier isomorphism) the set of decompositions of o/^1F\Lambda \Omega ffl X=S is an affine space under the same group (4.6(c)): One can show that the map Z 0 7! 'Z0 constructed in the proof of 5.1 is an affine bijection between these two spaces. (c) In fact, there is in [D-I] 3.5 a statement covering (a) and (b), by appealing to the theory gerbes of Giraud [Gi]. 2. The degeneration theorem 5.6 has a relative variant. We work under the assumption of 5.1, and denote by f : X ! S the structure morphism. Consider then the spectral sequence (4.8.1) relative to the functor f\Lambda and the complex \Omega ffl X=S , E ij 1 = R j f\Lambda \Omega i X=S ) R \Lambda f\Lambda (\Omega ffl X=S ); which is called the relative Hodge to de Rham spectral sequence (of X over S). Then if X is smooth and proper of relative dimension ! p, and if X 0 luc illusie, frobenius and hodge degeneration 126 is lifted over T , this spectral sequence degenerates at E1 and the sheaves R j f\Lambda \Omega i X=S are locally free of finite type. ([D-I] 4.1.5). 3. The latter assertion of 5.5 and the conclusions of 5.6 and 5.8 still remain true if one only assumes X of dimension ^ p ([D-1] 2.3). This is a consequence of Grothendieck duality for the morphism F . 4. There exists many examples of smooth and proper surfaces X over an algebraically closed field k of characteristic p for which the Hodge to de Rham spectral sequence does not degenerate at E1 and which does not satisfy the vanishing property of Kodaira-Akizuki-Nakano type of 5.8. (Taking into account (3) if p = 2, or 5.6 and 5.8 if p ? 2, these surfaces are therefore not lifted over W2(k).) See ([D-I] 2.6 and 2.10) for a bibliography on this subject. 5. Formulas (5.8.1) and (5.8.2) are still useful if d = 2 ^ p; X is liftable over W2(k) and L is only assumed numerically positive, i.e. satisfies L \Delta L ? 0 and L \Delta O(D) * 0 for any effective divisor D, see [D-I] 2. 6. From characteristic p ? 0p ? 0p ? 0 to characteristic zero 6.0. There exists a standard technique in algebraic geometry, which allows one to prove certain statements of geometric nature 15 , formulas over a base field of characteristic zero, from analogous statements over a field of characteristic p ? 0, even a finite field. Roughly speaking, it consists of a given base field K, which is in characteristic zero, as an inductive limit of its Z-sub-algebras of finite type Ai: Data on K, provided that they satisfy certain finiteness conditions, arise by extension of scalars from similar data on one of the Ai, say Ai 0 = B. It is then enough to solve the similar problem on T = Spec B, that which is seemingly more difficult. The advantage however, is that the closed points of T are then the spectrum of a finite field, and that in a sense which one can specify, there are many such points, so that it is enough to check the statement posed on T after sufficient specialization to these points. There is the business dealing with a problem of characteristic p ? 0, where one has the range of corresponding methods (Frobenius, Cartier isomorphism, etc.); moreover one can exploit the fact of being able to choose the characteristic large enough. The two ingredients of the method are: (a) results of passing to the limit, presented in great generality in (EGA IV 8), allowing the "spreading out" of certain data and properties on K, to similar data and properties on B; (b) density properties of closed points on schemes such that the schemes are of finite type over a field or over Z (EGA IV 10). 6.1. Let ((Ai)i2I ; uij : Ai ! Aj (i ^ j)) be a filtered inductive system of rings, with inductive limit A, and denote by ui : Ai ! A the canonical homomorphism. The two very important examples are: (i) a ring A written as an inductive limit of its sub-Z-algebras of finite type; (ii) the localization Ap of a ring A at a prime ideal p written as an inductive limit of localizations Af (= A[1=f]) for f =2 p. The prototype of problems and results of type (a) above is the following. Let (Ei) = ((Ei)i2I ; vij : Ei ! Ej ) be an inductive system of Ai-modules, having for inductive limit the A-module E. Let us agree to say that (Ei) is cartesian if, 15 I.e. stable by base extension, as opposed to statements of arithmetic nature, where the base plays an essential role. 127 6. from characteristic p ? 0 to characteristic zero for any i ^ j; vij (which is an Ai-linear homomorphism of Ei into Ej considered as an Ai-module via uij) induces, by adjunction, an isomorphism (Aj-linear) of u \Lambda ij Ei = Aj \Omega A i Ei in Ej . In this case, the canonical homomorphism vi : Ei ! E induces for all i, an isomorphism u \Lambda i Ei (= A \Omega A i Ei) , \Gamma ! E. Let ((Fi)i 2 I; wij) be a second inductive system of Ai-modules. If (Ei) is cartesian, the HomA i (Ei; Fi) form an inductive system of Ai-modules: The transition map for i ^ j associated to fi : Ei ! Fi is the homomorphism Ej ! Fj composed with the inverse of the isomorphism of Aj \Omega Ei in Ej defined by vij, from Aj \Omega fi : Aj \Omega Ei ! Aj \Omega Fi, and from the map of Aj \Omega Fi in Fj defined by wij. If F denotes the inductive limit of the Fi, one has analogous maps of HomA i (Ei; Fi) into HomA(E; F ), which defines a homomorphism (6.1.1) ind lim HomA i (Ei; Fi) ! HomA(E; F ): We can then pose the following two questions : (1) Being given an A-module E, does there exist i0 2 I and an Ai 0 -module Ei 0 such that E results from Ei 0 by an extension of scalars of Ai 0 to A (or, that which amounts to the same, does there exist a cartesian inductive system (Ei), indexed by fi 2 Iji * i0g, for which the limit is E) ? (2) If there exists i0 such that (Ei) and (Fi) are cartesian for i * i0, is the map (6.1.1) (where the inductive limit is reached for i * i0) an isomorphism ? There is a positive answer to the two questions with the help of hypothesis of finite presentation. (Recall that a module is said to be finitely presented if it is the cokernel of a homomorphism between free modules of the finite type.) More precisely, there is the following statement, which can be verified immediately: Lemma 6.1.2. With the preceding notation: (a) If E is a finitely presented A-module, there exists i0 2 I and an Ai 0 -module of finite presentation Ei 0 such that u \Lambda i0 Ei 0 ' E. (b) Let (Ei); (Fi) be two inductive systems, cartesian for i * i0, with respective inductive limits E and F . Then if Ei 0 is finitely presented, the map (6.1.1) is an isomorphism. It follows from this that if E is finitely presented, the Ei 0 which arises by extension of scalars is essentially unique, in this sense that if Ei 1 is another choice (Ei 0 and Ei 1 both being two finite presentations), there exists i2 with i2 * i1 and i2 * i0 such that Ei 0 and Ei 1 become isomorphisms by extensions of scalars to Ai 2 . The Si = Spec Ai form a projective system of schemes for which S = Spec A is the projective limit. If (Xi; vij : Xj ! Xi) is a projective system of Si-schemes, we say that this system is cartesian for i * i0 if, for i0 ^ i ^ j, the transition arrow vij gives a cartesian square Xj ! Xi # # Sj ! Si: In this case, the S-scheme induced from Xi 0 by extension of scalars to S is the projective limit of Xi. If (Yi) is a second projective system of Si-schemes, cartesian for i * i0, the projective limit Y (= S \Theta S i0 Yi 0 ) of the HomS i (Xi; Yi) form a luc illusie, frobenius and hodge degeneration 128 projective system, and one has an analogous map to (6.1.1): (6.1.3) proj lim HomS i (Xi; Yi) ! HomS(X; Y ): We can then formulate similar questions to (1) and (2) above. They have similar answers, with the condition of replacing the hypothesis of finite presentation for modules by the hypothesis of finite presentation for schemes (a morphism of schemes X ! Y is said to be a finite presentation if it is locally of finite presentation (2.1) and "quasi-compact and quasi-separated", which means that X is a finite union of open affine subsets Uff over an open affine subset Vff of Y and that the intersections Uff " Ufi have the same property; if Y is Noetherian, X is finitely presented over Y if and only if X is of finite type over Y , i.e. locally of finite type over Y (2.1) and Noetherian): Proposition 6.2. (a) If X is an S-scheme of finite presentation, there exists i0 2 I and an Si 0 -scheme Xi 0 of finite presentation for which X is induced by a base change. (b) If (Xi); (Yi) are two projective systems of Si-schemes, cartesian for i * i0, and if Xi 0 and Yi 0 are finitely presented over Si 0 , then the map (6:1:3) is bijective. As in the preceding, it follows from this that Xi 0 of 6.2 (a) is essentially unique (two such schemes become Si-isomorphic for i large enough). Moreover, the usual properties of an S-scheme of finite presentation (or of a morphism between such) are already determined to some extent, over Si for i large enough. Here are some, which are useful statements in themselves (the reader will find a long list in (EGA IV 8, 11.2, 17.7)): Proposition 6.3. Let X be an S-scheme of finite presentation. We assume that X has one of the following properties P: projective, proper, smooth. Then there exists i0 2 I and an Si 0 -scheme Xi 0 of finite presentation, having the same property P, for which X is induced by base change. The case where P is "projective" is easy: X is the closed subscheme of a standard projective space P = P r S defined by an ideal locally of finite type. It suffices to lift P , and then the closed immersion (i.e. the corresponding quotient of OP , cf. 6.11). The "proper" case is less immediate, but roughly, it goes back to a classical result, namely Chow's Lemma (cf. EGA IV 8.10.5). The "smooth" case is a little more difficult (which uses criterion 2.10), see (EGA IV 11.2.6 and 17.7.8). With regard to the properties of type (b) evoked in 6.0, we will only have need of the following result: Proposition 6.4. Let S be a scheme of finite type over Z. Then: (a) If x is a closed point of S, the residue field k(x) is a finite field, (b) All locally closed nonempty components Z of S contain a closed point of S. For the proof, we refer to (EGA IV 10.4.6, 10.4.7), or in the case where S is affine, this goes back to (Bourbaki, Alg. Com. V, by 3, n ffi 4) (this is a consequence of Hilbert's theorem of zeros). We will need to apply 6.4 (b) to the case where Z is the smooth part of S; S being assumed integral 16 : 16 A scheme is said to be integral if it is reduced and irreducible. 129 6. from characteristic p ? 0 to characteristic zero Proposition 6.5. Let S be an integral scheme of finite type over Z. The set of points x of S for which S is smooth over Spec Z is a nonempty open set of S. In particular, if A is a Z-algebra of finite type, and integral, there exists s 2 A; s 6= 0, such that Spec As is smooth over Z. The openness of the set of smooth points of a morphism locally of finite presentation is a general fact, which is a consequence for example of the jacobi criterion 2.6 (a), cf. (EGA IV 12.1.6.). That in the present case this open set is nonempty follows from a local variant of 2.10 and from the fact that the generic fiber of S is smooth over Q at its generic point, Q being perfect. We will finally have to use some standard results of compatability of direct images by a base change (or, as one says sometimes, of cohomological cleanliness). Not wanting to weigh down our exposition, we will state them only in the case where it will be useful for us to have, for the Hodge cohomology and the de Rham cohomology. Proposition 6.6. Let S be an affine scheme 17 , Noetherian, integral, and f : X ! S a smooth and proper morphism. (a) The sheaves R j f\Lambda \Omega i X=S and R n f\Lambda \Omega ffl X=S are coherent. There exists a nonempty open set U of S such that, for any (i; j) and any n, the restrictions to U of these sheaves are locally free of finite type. (b) For any i 2 Z and for any morphism g : S 0 ! S, if f 0 : X 0 ! S 0 denotes the induced scheme of X by base change via g, the canonical arrows of D(S 0 ) (according to base change) Lg \Lambda Rf\Lambda \Omega i X=S ! Rf 0 \Lambda \Omega i X0=S0 (6.6.1) Lg \Lambda Rf\Lambda \Omega ffl X=S ! Rf 0 \Lambda \Omega ffl X0=S0 (6.6.2) are isomorphisms. (c) Fix i 2 Z and assume that for any j, the sheaf R j f\Lambda \Omega i X=S is locally free over S, of constant rank h ij . Then for any j, the base change arrow (induced from (6.6.1)) (6.6.3) g \Lambda R j f\Lambda \Omega i X=S ! R j f 0 \Lambda \Omega i X0=S0 is an isomorphism. In particular, R j f 0 \Lambda \Omega i X0=S0 is locally free of rank h ij . (d) Suppose that for all n, R n f\Lambda \Omega ffl X=S is locally free of constant rank hn. Then for all n, the change of base arrow (induced from (6.62)) (6.6.4) g \Lambda R n f\Lambda \Omega ffl X=S ! R n f 0 \Lambda \Omega ffl X0=S0 is an isomorphism. In particular, R n f 0 \Lambda \Omega ffl X0=S0 is locally free of rank h n . Let us briefly indicate the proof. The fact that the R j f\Lambda \Omega i X=S are coherent is a particular case of the finiteness theorem of Grothendieck (EGA III 3) (or [H2] III 8.8 in the projective case). The coherence of R n f\Lambda \Omega ffl X=S follows from this by the relative Hodge to de Rham spectral sequence (5.9(2)). For the second 17 The hypothesis "affine" is unnecessary; we use it only to facilitate the proof of (b). luc illusie, frobenius and hodge degeneration 130 assertion of (a), denote by A the (integral) ring of S; K its field of fractions, which is therefore the local ring of S at its generic point j. We set for abbreviation R j f\Lambda \Omega i X=S = H ij ; R n f\Lambda \Omega ffl X=S = H n . The fiber of H ij (resp. H n ) at j is free of finite type (a K-vector space of finite dimension), and is the inductive limit of H ij jD(s) (resp. H n jD(s) ), for s transversing A; D(s) denoting "the open complement" of s, i.e. Spec As = X \Gamma V (s). By 6.1.2 it follows from this that there exists s such that H ij jD(s) (resp. H n jD(s) ) are free of finite type. For (b), we choose a finite covering U of X by open affine sets, denote by U 0 the open covering of X 0 induced from U by base change. Since S is affine and that X is proper, therefore separated over S, the finite intersections of open sets in U are affine and similarly the finite intersections of open sets in U 0 are (relatively) affine 18 over S 0 . Consequently (cf. [H2] III 8.7), Rf\Lambda \Omega i X=S (resp. Rf 0 \Lambda \Omega i X0=S0 ) is represented by f\Lambda ^ C(U; \Omega i X=S ) (resp. f 0 \Lambda ^ C(U 0 ; \Omega i X0=S0 )), where we denote here by ^ C(U; ffl) the alternating complex of cochains. By the compatability of \Omega i by a base change, there is a canonical isomorphism of complexes g \Lambda f\Lambda ^ C(U; \Omega i X=S ) , \Gamma ! f 0 \Lambda ^ C(U 0 ; \Omega i X0=S0 ): Since the complex f\Lambda ^ C(U; \Omega i X=S ) is bounded and with flat components, this isomorphism realizes the isomorphism (6.6.1). Similarly, Rf\Lambda \Omega ffl X=S (resp. Rf 0 \Lambda \Omega ffl X0=S0 )) is represented by f\Lambda ^ C(U; \Omega ffl X=S ) (resp. f 0 \Lambda ^ C(U 0 ; \Omega ffl X0=S0 )) (where ^ C denotes this time the associated simple complex of the ^ Cech bicomplex), and one has a canonical isomorphism of complexes g \Lambda f\Lambda ^ C(U; \Omega ffl X=S ) , \Gamma ! f 0 \Lambda ^ C(U 0 ; \Omega ffl X0=S0 ); which realizes the isomorphism (6.6.2). Assertions (c) and (d) follow from (b) and from the following lemma, for which we leave the verification to the reader: Lemma 6.7. Let A be a Noetherian ring and E a complex of A-modules such that H i (E) are projective of finite type for any i and zero for almost all i. Then: (a) E is isomorphic, in D(A), to a bounded complex with projective components of finite type. (b) If E is bounded and with projective components of finite type, for any A-algebra B, and for all i, the canonical homomorphism B \Omega A H i (E) ! H i (B \Omega A E) is an isomorphism. Remarks 6.8. (a) A complex of A-modules, isomorphic in D(A), to a bounded complex with projective components of finite type is said to be perfect. One must be aware that if E is perfect, it is not true in general, that the H i (E) are projective of finite type. One can show that under the hypothesis of 6.6, the complexes Rf\Lambda \Omega i X=S and Rf\Lambda \Omega ffl X=S are perfect over S (and not only over U ). The notion of a perfect complex plays an important role in numerous questions in algebraic geometry. (b) In the statements of 6.6 concerning \Omega i X=S , one can replace \Omega i X=S by any locally free OX -module F of finite type (even coherent and relatively flat over S): The 18 A morphism of schemes is said to be affine if the inverse image of any affine open set is affine. 131 6. from characteristic p ? 0 to characteristic zero conclusions of (a), (b) and (c) are still valid on the condition of replacing \Omega i X0=S0 by the inverse image sheaf F 0 of F over X 0 . Similarly, the complex Rf\Lambda F is perfect over S. We are now able to state and prove the promised application of 5.6: Theorem 6.9 (Hodge Degeneration Theorem). Let K be a field of characteristic zero, and X a smooth and proper K-scheme. Then the Hodge spectral sequence of X over K (4.8.3) E ij 1 = H j (X; \Omega i X=K ) ) H \Lambda DR (X=K) degenerates at E1. Set dimK H j (X; \Omega i X=K ) = h ij ; dim H n DR (X=K) = h n . It suffices to prove that for all n; h n = P i+j=n h ij (cf. (4.8.3)). Write K as an inductive limit of the family (A*)*2L of its sub-Z-algebras of finite type. According to 6.3, there exists ff 2 L and a smooth and proper Sff-scheme Xff (where Sff = Spec Aff) for which X is induced by base change Spec K ! Sff. Even if it means to replace Aff by Aff[t \Gamma 1 ] for a suitable nonzero t 2 Aff, we can assume, according to 6.5, that Sff is smooth over Spec Z. Abbreviate Aff by A, Sff by S; Xff by X, and denote by f : X ! S the structure morphism. Again by replacing A by A[t \Gamma 1 ], we can according to 6.6 (a), assume that the sheaves R j f\Lambda \Omega i X=S (resp. R n f\Lambda \Omega ffl X=S ) are free of constant rank, necessarily equal then to h ij (resp. h n ) according to 6.6 (c) and (d). Since the relative dimension of X over S is a locally constant function and that X is quasicompact, one can in addition choose an integer d which bounds this dimension at any point of X and therefore the dimension of the fibers of X over S at any point of S. Applying 6.4 (b) to Z = Spec A[1=N ] for suitable N (say, the product of prime numbers ^ d), one can choose a closed point s of S, for which the residue field k = k(s) (a finite field) is of characteristic p ? d. Since S is smooth over Spec Z, the canonical morphism Spec k ! S (a closed immersion) is extended (by definition of smoothness (2.2)) to a morphism g : Spec W2(k) ! S, where W2(k) is the ring of Witt vectors of length 2 over k (3.9). Denote by Y = Xs the fiber of X over s = Spec k and Y1 the scheme over Spec W2(k) induced from X by the base change g. We therefore have cartesian squares: Y ! Y1 ! X X # # # # s ! Spec W2(k) g \Gamma ! S Spec K: By construction, Y is a smooth and proper k-scheme of dimension ! p, lifted over W2(k). Therefore according to 5.6, the Hodge to de Rham spectral sequence of Y over k degenerates at E1. We therefore have for all n, X i+j=n dimk H j (Y; \Omega i Y=k ) = dimk H n DR (Y=k): But according to 6.6 (c) and (d), we have for all (i; j) and for all n, dimk H j (Y; \Omega i Y=k ) = h ij ; dimk H n DR (Y=k) = h n : Therefore P i+j=n h ij = h n for all n, for which the proof follows. luc illusie, frobenius and hodge degeneration 132 Theorem 6.10 (Kodaira-Akizuki-Nakano Vanishing Theorem [KAN], [AkN]). Let K be a field of characteristic zero, X a smooth projective K-scheme of pure dimension d, and L an ample invertible sheaf on X. Then we have: H j (X; L \Omega \Omega i X=K ) = 0 for i + j ? d;(6.10.1) H j (X; L \Omega \Gamma 1 \Omega \Omega i X=K ) = 0 for i + j ! d:(6.10.2) We deduce 6.10 from 5.8 just like 6.9 from 5.6. We need for this a result of passing to the limit for modules, generalizing and clarifying 6.1.2 (cf. (EGA IV 8.5, 8.10.5.2)): Proposition 6.11. Assume given, as in 6.1, a filtered projective system of affine schemes (Si)i2I , with limit S. Let i0 2 I; Xi 0 be a Si 0 -scheme of finite presentation and consider the induced projective cartesian system (Xi) for i * i0, with limit X = S \Theta S i0 Xi 0 . (a) If E is a finitely presented OX -module, there exists i * i0 and a OX i -module Ei of finite presentation for which E is induced by extension of scalars. If E is locally free (resp. locally free of rank r), there exists j * i such that Ej = OX j \Omega O Xi Ei is locally free (resp. locally free of rank r). If X is projective over S and E is an ample invertible OX -module (resp. very ample) (5.7), there exists j * i such that Xj is projective over Sj and Ej is ample invertible (resp. very ample). (b) Let Ei 0 ; Fi 0 be finitely presented OX -modules, and consider the systems (Ei); (Fi) which are induced by extension of scalars over the Xi for i * i0, as well as the modules E and F which are induced by extension of scalars over X. Then there is a natural map ind lim i*i0 HomO Xi (Ei; Fi) ! HomO X (E; F ); which is bijective. The proof of (b), then of the first two assertions of (a), brings us back to 6.1.2. For the latter part of (a), it suffices treat the case where E is very ample, i.e. corresponds to a closed immersion h : X ! P = P r S such that h \Lambda OP (1) ' E. For i sufficiently large, one lifts h by an Si-morphism hi : Xi ! Pi = P r Si and E by invertible Ei over Xi. Even if it means to increase i, hi is a closed immersion and the isomorphism h \Lambda OP (1) ' E comes from an isomorphism h \Lambda i OP i (1) ' Ei; Ei is then very ample. Proving 6.10. Proceeding as in the proof of 6.9, and moreover applying 6.11, one can find a subring A of K of finite type and smooth over Z, a smooth projective morphism f : X ! S = Spec A of pure relative dimension d, for which X ! Spec K is induced by base change, and an ample invertible OX -module L for which L is induced by extension of scalars. By virtue of 6.6 and 6.8 (b), one can assume, even if it means to replace A by A[t \Gamma 1 ], that the sheaves R j f\Lambda (M\Omega \Omega i X =S), where M = L (resp. L \Omega \Gamma 1 ), are free of finite type, of constant rank, necessarily equal, according to 6.8 (b), to h ij (L) = dimK H j (X; L \Omega \Omega i X=K ) (resp. h ij (L \Omega \Gamma 1 ) = H j (X; L \Omega \Gamma 1 \Omega \Omega i X=K )). Let us choose then g : Spec W2(k) ! S as in the proof of 6.9. The inverse image sheaf Ls of L over Y = Xs is ample. According to 6.6 and 6.8 (b), one has dimk H j (Y; Ls \Omega \Omega i Y=k ) = h ij (L), and dimk H j (Y; L \Omega \Gamma 1 s \Omega \Omega i Y=k ) = h ij (L \Omega \Gamma 1 ). The conclusion then follows from 5.8. 133 7. recent developments and open problems Remark 6.12. In a similar manner, the Ramanujam vanishing theorem on surfaces [Ram] follows from the variant of 5.8 relative to the numerically positive sheaves (cf. 5.9 (5)). 7. Recent developments and open problems A. Divisors with normal crossings, semi-stable reduction, and logarithmic structures. 7.1. Let S be a scheme, X a smooth S-scheme, and D a closed subscheme of X. We say that D is a divisor with normal crossings relative to S (or simply, relative) if, "locally for the 'etale topology on X", the couple (X; D) is "isomorphic" to the couple formed from the standard affine space A n S = S[t1; : : : ; tn] and from the divisor V (t1 \Delta \Delta \Delta tr) of the equation t1 \Delta \Delta \Delta tr = 0, for 0 ^ r ^ n (the case r = 0 corresponds to t1 \Delta \Delta \Delta tr = 1 and V (t1 \Delta \Delta \Delta tr) = ;). This means that there exists an 'etale covering (Xi)i2I of X (i.e. a family of 'etale morphisms Xi ! X for which the union of the images is X) such that, if Di = Xi \Theta X D is the closed subscheme induced by D on Xi, there exists an 'etale morphism Xi ! A n S for which there is a cartesian square Di ! Xi # # V (t1 \Delta \Delta \Delta tr) ! A n S (n and r dependant on i). In other words, that there exists a coordinate system (x1; : : : ; xn) on Xi in the sense of 2.7 (defining the 'etale morphism Xi ! A n S ) such that Di is the closed subscheme of the equation x1 \Delta \Delta \Delta xr = 0. This definition is modeled after the analogous definition in complex analytic geometry (cf. [D1]), where "locally for the 'etale topology" is replaced by "locally for the classical topology", and "'etale morphism" by "local isomorphism". A standard example of a divisor with normal crossings relative to S = Spec k; k a field of characteristic different from 2, is the cubic with double point D = Spec k[x; y]=(y 2 \Gamma x 2 (x \Gamma 1)) in the affine plane X = Spec k[x; y]. (Observe in this example that there does not exist a system of coordinates (xj ) as above on a Zariski open covering of X, an 'etale extension (extraction of a square root of x \Gamma 1) being necessary for to make possible such a system in a neighbourhood of the origin.) The notion of a divisor with the normal crossings D ,! X relative to S is stable by 'etale localization over X and by base change S 0 ! S. If D ,! X is a relative divisor with normal crossings, and if j : U = XnD ,! X is the inclusion of the open complement, we define a subcomplex (7.1.1) \Omega ffl X=S (log D) of j\Lambda \Omega ffl U=S , called the de Rham complex of X=S with logarithmic poles along D, by the condition that a local section ! of j\Lambda \Omega i U=S belong to \Omega i X=S (log D) if and only if ! and d! have at most a simple pole along D (i.e. are such that if f is a local equation of D; f ! (resp. f d!) is a section of \Omega i X=S (resp. \Omega i+1 X=S ) (NB. f is necessarily a nonzero divisor in OX )). One easily sees that the OX -modules \Omega i X=S (log D) are locally free of finite type, that \Omega i X=S (log D) = \Lambda i \Omega 1 X=S (log D), and that if as above, (x1; : : : ; xn) are coordinates on an X 0 'etale neighbourhood luc illusie, frobenius and hodge degeneration 134 over X where D has for equation x1 \Delta \Delta \Delta xr = 0; \Omega 1 X=S (log D) is free with basis (dx1=x1; : : : ; dxr=xr; dxr+1; : : : ; dxn). There is a natural variant in complex analytic geometry of the construction (7.1.1) (cf. [D1]). If S = Spec C and D ae X is a(n algebraic) divisor with normal crossings, the complex of analytic sheaves associated to (7.1.1) on the analytic space X an associated to X, \Omega ffl X=C (log D) an = \Omega ffl Xan=C (log D an ); calculates the transcendental cohomology of U with values in C : There is a canonical isomorphism (in the derived category D(X an ; C )) (7.1.2) Rj\Lambda C ' \Omega ffl X=S (log D) an ; and consequently an isomorphism (7.1.3) H i (U an ; C ) ' H i (X an ; \Omega ffl X=S (log D) an ) (loc. cit.). Moreover, if X is proper over C , the comparison theorem of Serre [GAGA] allows us to deduce from (7.1.3) the isomorphism (7.1.4) H i (U an ; C ) ' H i (X; \Omega ffl X=S (log D)): Moreover the filtration F of H \Lambda (X; \Omega ffl X=S (log D)), being the outcome of the first spectral sequence of hypercohomology of X with values in \Omega ffl X=S (log D), (7.1.5) E pq 1 = H q (X; \Omega p X=S (log D) ) H p+q (X; \Omega ffl X=S (log D)) is the Hodge filtration of the natural mixed Hodge structure of H \Lambda (U an ; Z) defined by Deligne, and the spectral sequence (7.1.5) degenerates at E1 ([D2]). Just as in the case where D = ; (6.9), this degeneration can be shown by reduction to characteristic p ? 0. Indeed, we have the following result which generalizes 5.1 and for which the proof is analogous ([D-I] 4.2.3): Theorem 7.2. Let S be a scheme of characteristic p ? 0; S , a flat lifting of S over Z=p 2 Z; X a smooth S-scheme and D ae X a relative divisor with normal crossings. Denote by F : X ! X 0 the relative Frobenius of X=S. If the couple (X 0 ; D 0 ) admits a lifting (X 0, ; D 0, ) over S , , where X 0, is smooth and D 0, ae X 0, is a relative divisor with normal crossings, the complex of OX -modules o/!pF\Lambda \Omega ffl X=S (log D) is decomposable in the derived category D(X 0 ). The reader will find in [D-I] various complements to 7.2 and in [E-V] another presentation of the same results, and some applications of the theorems pertaining to ampleness and vanishing results. 7.3. The preceeding theory extends without much change to a class of morphisms which are no longer smooth, but not far from this, namely the morphisms that are said to be "of semi-stable reduction". Let T be a scheme. The prototype of such morphisms is the morphism s : A n T = T [x1; : : : ; xn] ! A 1 T = T [t]; t 7! x1 \Delta \Delta \Delta xn (n * 1): In other words, if S = A 1 T , the scheme A n T , considered as S-scheme by s, is the sub-S-scheme of A n S = S[x1; : : : ; xn] = T [x1; : : : ; xn; t] with equation x1 \Delta \Delta \Delta xn = t. 135 7. recent developments and open problems The morphism s is smooth outside 0 and its fiber at 0 is the divisor D with equation (x1 \Delta \Delta \Delta xn = 0), a divisor with normal crossings relative to T , but not with S (a "vertical" divisor). More generally, if S is a smooth T -scheme of relative dimension 1 and E ae S a relative divisor with normal crossings (if T is the spectrum of an algebraically closed field, E is therefore simply a finite set of rational points of S), we say that the S-scheme X has semi-stable reduction along E if, locally for the 'etale topology (over X and over S) the morphism X ! S is of the form s ffi g, with g smooth, s being the morphism considered above. The divisor D = X \Theta S E ae X is then a divisor with normal crossings relative to T (but not to S) 19 . An elementary example is furnished by the "Legendre family" X = Spec k[x; y; t]=(y 2 \Gamma x(x \Gamma 1)(x \Gamma t)) over S = Spec k[t], (k a field of characteristic 6= 2), which has semi-stable reduction on f0g [ f1g, the fiber at each of these points being isomorphic to the cubic with double point considered above. The interest in the notion of semi-stable reduction comes from the semi-stable reduction conjecture, which roughly asserts that locally, after suitable ramification of the base, a smooth morphism can be extended to a morphism with semi-stable reduction. This conjecture was established by Grothendieck-Deligne-Mumford and Artin-Winters ([G], [A-W], [D-M]) in any characteristic but relative dimension 1, and Mumford ([M]) in characteristic zero and arbitrary relative dimension. If f : X ! S has semi-stable reduction along E, we define the de Rham complex with relative logarithmic poles (7.3.1) ! ffl X=S = \Omega ffl X=S (log D=E); with components ! i X=S = \Lambda i ! 1 X=S , where ! 1 X=S is the quotient of \Omega 1 X=T (log D) by the image of f \Lambda \Omega 1 S=T (log E) and the differential is induced from that of \Omega ffl X=T (log D) by passing to the quotient. This complex has locally free components of finite type (in the case of the morphism s above, ! 1 X=S is isomorphic to ( L OX dxi=xi)=OX ( P dxi=xi) (therefore free with basis dxi=xi; i * 2)). It induces on the smooth open part U of X over S the usual de Rham complex \Omega ffl U=S , and one can show that this is the unique extension over X of this complex which has locally free components of finite type. Moreover, if one sets for abbreviation, ! ffl X=T = \Omega ffl X=T (log D); ! ffl S=T = \Omega ffl S=T (log E), there is an exact sequence (7.3.2) 0 ! ! 1 S=T \Omega ! ffl X=S [\Gamma 1] ! ! ffl X=T ! ! ffl X=S ! 0; where the arrow to the left is given by a\Omega b 7! f \Lambda a^b. This exact sequence plays an important role in the regularity theorem of the Gauss-Manin connection (cf. [K2] and the article of Bertin-Peters in this volume). There also exists a variant of these constructions in complex analytic geometry. Assume that T = Spec C , that S is a smooth curve over C ; E ae S the divisor reduced to a point 0, and that f : X ! S is a morphism with semi-stable reduction at f0g, with fiber Y at 0. (Y is therefore a divisor with normal crossings in X relative to C .) We consider the complex (7.3.3) ! ffl Y = C f0g \Omega O S ! ffl X=S ; with components the locally free sheaves of finite type ! i Y = OY \Omega O X ! i X=S . Steenbrink [St] has shown that the complex analogue ! ffl Y an over Y an (which is also the 19 One can similarly define a notion of semi-stable reduction along E without the hypothesis on the relative dimension of S over T , cf. [I5]. luc illusie, frobenius and hodge degeneration 136 complex of sheaves associated to ! ffl Y over Y an ) embodies the complex of neighbouring cycles R\Psi (C ) of f at 0, so that if moreover f is proper, H \Lambda (Y; ! ffl Y ) "calculates" H \Lambda (X an t ; C ) for t "close enough" to 0. Steenbrink also shows (under this extra hypothesis) that the spectral sequences (7.3.4) E pq 1 = R q f\Lambda ! p X=S ) R p+q f\Lambda ! ffl X=S and (7.3.5) E pq 1 = H q (Y; ! p Y ) ) H p+q (Y; ! ffl Y ) degenerate at E1 and that the sheaves R q f\Lambda ! p X=S are locally free of finite type and of formation compatible with any base change. These results form part of the construction of a limiting mixed Hodge structure on H \Lambda (X an t ; Z) for t tending to 0 (loc. cit.). They can by themselves, be proven by reduction to characteristic p ? 0 ([I5]). For T of characteristic p ? 0, and f : X ! S with semi-stable reduction along E ae S, the complexes ! ffl X=S and (7.3.6) ! ffl D = OD \Omega O S ! ffl X=S (where D = E \Theta S X) indeed give rise to Cartier morphisms (of the type of 3.5), and under the hypothesis of a suitable lifting modulo p 2 , o/!pF\Lambda ! ffl X=S and o/!pF\Lambda ! ffl D decompose (in D(X 0 )). (See [I5] 2.2 for a precise statement, which generalizes 7.2 and other corollaries (degeneration and vanishing statements).) 7.4. The complex ! ffl D above does not depend only on D, but on X=S. It does depend on it however locally (in a neighbourhood of D). While seeking to elucidate the additional structure on D necessary for the definition, J.-M. Fontaine and the author were led to introduce the notion of logarithmic structure. This paved the way to a theory, logarithmic geometry, as a natural extension of the theory of schemes. Widely developed by K. Kato and his school, it makes possible to unify the various constructions of complexes with logarithmic poles considered above and to consider the toric varieties of Mumford et al. and the morphisms with semi-stable reduction as particular cases of a novel notion of smoothness. See [I6] for an introduction. The preceding decomposition, degeneration and vanishing results admit generalizations in this program, see [Ka2] and [Og2]. B. Degeneration mod p n and crystals. 7.5. The decomposition Theorem 5.1 was originally obtained as a by-product of the work of Ogus [Og1], Fontaine-Messing [F-M] and Kato [Ka1] by crystalline cohomology (see [I4] for a panorama of this theory). The link (a small technique) between 5.1 and the point of view of crystalline is explicit in [D-I] 2.2 (iv). We limit ourselves to a statement of a degeneration result mod p n ([F-M], [Ka1]) analogous to 5.6: Theorem 7.6. Let k be a perfect field of characteristic p ? 0; W = W (k) the ring of Witt vectors over k; X a smooth and proper W -scheme of relative dimension ! p. Then for any integer n * 1, the Hodge to de Rham spectral sequence (7.6.1) E ij 1 = H j (Xn; \Omega i Xn=Wn ) ) H i+j DR (Xn=Wn) 137 7. recent developments and open problems degenerates at E1, where Wn = Wn(k) = W=p n W denotes the ring of Witt vectors of length n over k and Xn the scheme over Wn induced from X by reduction modulo p n (i.e. by extension of scalars of W to Wn). 7.7. For n = 1, we have Wn = Wn(k) = W=p n W and we recover statement 5.6, apart from which in 7.6, we assume given a lifting of X over W (rather than over W2) 20 . Under the hypothesis of 7.6, it is not true in general, for n * 2, that the de Rham complex \Omega ffl Xn=Wn (which is, a priori, only a complex of sheaves of Wn-modules over Xn (or X1, Xn and X1 having the same underlying space)) is decomposable in the corresponding derived category D(X1; Wn). However, the results of Ogus ([Og1] 8.20) imply that if oe denotes the Frobenius automorphism of Wn; oe \Lambda \Omega ffl Xn=Wn is isomorphic in the derived category D(X1; Wn) of sheaves of Wn-modules over X1, to the complex \Omega ffl Xn;Wn (p) induced from \Omega ffl Xn;Wn by multiplying the differential by p. (NB. For n = 1, we have \Omega ffl Xn=Wn (p) = L \Omega i X1=k [\Gamma i].) The conclusion of 7.6 comes about easily, like various additional properties of H \Lambda DR (Xn=Wn) (structure called "of Fontaine-Laffaille" - including in particular the fact that the Hodge filtration is formed from direct factors), see [F-M] and [Ka1]. 7.8. The degeneration and decomposition results for which we discussed until now carry over to de Rham complexes of schemes, possibly with logarithmic poles. More generally, we can consider the de Rham complexes with coefficients in modules with integrable connections. Many generalizations of this type have been obtained: For Gauss-Manin coefficients [I5], of sheaves of Fontaine-Laffaille [Fa2], of T -crystals [Og2] (besides these last objects providing a common generalization of the previous two). C. Open problems. 7.9. Let k be a perfect field of characteristic p ? 0; X a smooth k-scheme of dimension d; X 0 the scheme induced from X by base change by the Frobenius automorphism of k; F : X ! X 0 the relative Frobenius (3.1). We have seen in 5.9 (1) (a) (with S = Spec k; T = Spec W2(k)) that the following conditions are equivalent : (i) X 0 - or, that which amounts to the same here, X - is lifted (by a smooth and proper scheme) over W2(k); (ii) o/^1F\Lambda \Omega ffl X=k is decomposable in D(X 0 ) (4.6); (iii) o/!pF\Lambda \Omega ffl X=k is decomposable in D(X 0 ). We say that X is DR-decomposable if F\Lambda \Omega ffl X=k is decomposable (in D(X 0 )). As we have observed in 5.2, this condition is equivalent, taking into account the Cartier isomorphism (3.5), to the existence of an isomorphism M \Omega i X0=k [\Gamma i] , \Gamma ! F\Lambda \Omega ffl X=k of D(X 0 ) inducing C \Gamma 1 on H i . The arguments of 5.6 and 5.8 show that: 20 In fact, Ogus has shown - albeit more difficult - that being given n * 1 and Z smooth and proper over Wn of dimension ! p, then if Z admits a lifting (smooth and proper) over Wn+1, the Hodge to de Rham spectral sequence of Z=Wn degenerates at E1 ([Og2] 8.2.6). This result truly generalizes 5.6. luc illusie, frobenius and hodge degeneration 138 (a) If X is proper over k and DR-decomposable, the Hodge to de Rham spectral sequence of X=k degenerates at E1. (b) If X is projective over k, of pure dimension d, and DR-decomposable, and if L is an invertible ample sheaf over X, one has the vanishing results of KodairaAkizuki-Nakano (5.8.1) and (5.8.2). By virtue of the equivalence between conditions (i) and (ii) above, a necessary condition for that X is DR-decomposable is that X is lifted over W2(k). According to [D-I], it is sufficient if d ^ p (5.5 and 5.9 (3)). We are unaware if it is always true in general: Problem 7.10. Let X be a smooth k-scheme of dimension d ? p, liftable over W2(k). Is it the case that X is DR-decomposable ? 7.11. Recall (5.5.1) that if X and F lift over W2(k); X is DR-decomposable; this is the case if X is affine, or is a projective space over k. As indicated in [D-I] 2.6 (iv), if X is liftable over W2(k) and if, for any integer n * 1, the product morphism (\Omega 1 X=k ) \Omega n ! \Omega n X=k admits a section, then X is DR-decomposable (see 8.1 for a proof). This second condition is checked in particular if X is parallelizable, i.e. if \Omega 1 X=k is a free OX-module (or, that which amounts to the same, the tangent bundle TX=k, dual of \Omega 1 X=k , is trivial), therefore for example if X is an abelian variety. By a theorem of Grothendieck (cf. [Oo] and [I7] Appendix 1), any abelian variety over k is lifted over W2(k) (and similarly over W (k)). Therefore any abelian variety over k is DR-decomposable. Another interesting class of liftable k-schemes (over W (k)) is formed from complete intersections in P r k (see the expose of Deligne (SGA 7 XI) for the definitions and basic properties of these objects). But we do not know if those are DR-decomposable. The first unknown case is that of a (smooth) quadric of dimension 3 in characteristic 2. We also don't know if the Grassmannians, and more generally, flag varieties, which are, albeit liftable over W (k), are DR-decomposable (the only known example is projective space!). Problem 7.10, with "liftable over W2(k)" replaced by "liftable over W (k)", is also an open problem. On the other hand, we can replace "liftable over W (k)" by "liftable over A", where A is a totally ramified extension of W (k) ( = ring of complete discrete valuations, finite and flat over W (k), with residue field k, and of degree ? 1 over W (k)): Lang [L] has indeed constructed in any characteristic p ? 0, a smooth projective k-surface X liftable over such a ring A of degree 2 over W (k) such that the Hodge to de Rham spectral sequence of X=k does not degenerate at E1. 7.12. The decomposition statements to which we referred to at the end of 7.3 apply in particular to a smooth curve S over T = Spec k and with a scheme X over S having semi-stable reduction along a divisor with normal crossings E ae S (therefore 'etale over k), for which certain hypothesis of liftability modulo p 2 are satisfied. More precisely, if we assume that: (i) There exists a lifting (E , ae S , ) of (E ae S) over W2 = W2(k) (with S , smooth and E , a relative divisor with normal crossings, i.e. 'etale over W2), admitting a lifting F , : S , ! S , of the Frobenius (absolute) of S such that (F , ) \Gamma 1 (E , ) = pE , 21 , 21 This notation denotes the divisor induced from E , by the raising to the p-th power of its local equations. 139 8. appendix: parallelizability and ordinary (ii) f is lifted by f , : X , ! S , having semi-stable reduction along E , , then o/!pF\Lambda ! ffl X=S and o/!pF\Lambda ! ffl D (where D = E \Theta S X) are decomposable, (and therefore F\Lambda ! ffl X=S and F\Lambda ! ffl D are also if X is of relative dimension ! p over S). 7.13. The relative result of ! ffl D suggests the following problem of a different characteristic. Now denote by S the spectrum of W = W (k), and E = Spec k the closed point of S. Let X be an S-scheme. By analogy with the definition given in 7.3, we say that X has semi-stable reduction if, locally in the 'etale topology (on X and on S), X is smooth over the subscheme of A n S = S[x1; : : : ; xn] with equation x1 \Delta \Delta \Delta xn = p. Assume that X has semi-stable reduction. Then X is a regular scheme, its fiber XK at the generic point Spec K of S (K = the field of fractions of W ) is smooth, and its fiber D = E \Theta S X at the closed point is a "divisor with normal crossings" in X. In this situation, we define a complex ! ffl X=S analogous to (7.3.1), which is the unique extension, with locally free components of a finite type of the de Rham complex of U over S, where U is the smooth open part of X over S. If X = S[x1; : : : ; xn]=(x1 \Delta \Delta \Delta xn \Gamma p), the OX-module ! 1 X=S , considered as a subsheaf of \Omega 1 XK=K , is identified with ( L OXdxi=xi)=OX ( P dxi=xi). The complex ! ffl D , defined by the formula (7.3.6), has locally free components of finite type over D, and coincides with the de Rham complex \Omega ffl D=k over the smooth open part of D. One has ! i X=S = \Lambda i ! 1 X=S and ! i D = \Lambda i ! 1 D . According to a result of Hyodo [Hy] (generalized by Kato in [Ka2]), the complex ! ffl D gives rise to a Cartier isomorphism C \Gamma 1 : ! i D0 ' H i F\Lambda ! ffl D . The complexes ! ffl X=S and ! ffl D play an important role in the recent developments of the theory of p-adic periods (cf. [I4] for a general view). Problem 7.14. With the notation of 7.13, let X be a semi-stable S-scheme with fiber D at the closed point of S. Is it the case that o/!pF\Lambda ! ffl D is decomposable (in D(D 0 )) ? Note that the decomposability of o/!pF\Lambda ! ffl D is equivalent to that of o/^1F\Lambda ! ffl D (same argument as in step C of the proof of 5.1). The answer is yes according to 5.5 if X is smooth over S. (NB. What was called X (resp. S) in 5.5 is here D (resp. E).) The answer is still yes (for trivial cohomological reasons (cf. 4.6 (a))) if X is affine, or if X is of relative dimension ^ 1. But the general case is unknown. 7.15. Finally, with regard to the vanishing theorem, we cannot prove by the methods of characteristic p ? 0, the classical results of Grauert-Riemenschneider or Kawamata-Viehmeg. Neither can we generalize 5.9 (5) in dimension ? 2. See [E-V] for a discussion of these questions. 8. Appendix: parallelizability and ordinary In this section, k denotes a perfect field of characteristic p ? 0. We denote by Wn = Wn(k) the ring of Witt vectors of length n over k. We begin by giving a proof of the result mentioned in 7.11: Proposition 8.1. Let X be a smooth k-scheme. Assume that X lifts over W2 and that for any n * 1, the product morphism (\Omega 1 X=k ) \Omega n ! \Omega n X=k admits a section. Then X is DR-decomposable (7.9). We will have need of the following lemma: luc illusie, frobenius and hodge degeneration 140 Lemma 8.2. Let S and T be as in 5:1; X a smooth scheme over S; Z 0 a (smooth) lifting of X 0 over T . Let ' 1 : \Omega 1 X0=S [\Gamma 1] ! F\Lambda \Omega ffl X=S be the homomorphism ' 1 Z0 of D(X 0 ) defined in step B of the proof of 5:1, and for n * 1, n : (\Omega 1 X0=S ) \Omega n [\Gamma n] ! F\Lambda \Omega ffl X=S the composite homomorphism ss ffi (' 1 ) L \Omega n , where ss : (F\Lambda \Omega ffl X=S ) L \Omega n ! F\Lambda \Omega ffl X=S is the product homomorphism. Likewise, we denote by ss : (\Omega 1 X0=S ) \Omega n ! \Omega n X0=S the product homomorphism. Then for any local section ! of (\Omega 1 X0=S ) \Omega n , one has H n n (!) = C \Gamma 1 ffi ss(!); where C \Gamma 1 : \Omega n X0=S ! H n F\Lambda \Omega ffl X=S is the Cartier isomorphism. Proof. It suffices to show this for ! of the form !1 \Omega \Delta \Delta \Delta \Omega !n, where !i is a local section of \Omega 1 X0=S . By functoriality in the Ei 2 D(X 0 ), of the product H 1 E1 \Omega \Delta \Delta \Delta \Omega H 1 En ! H n (E1 L \Omega \Delta \Delta \Delta L \Omega En); a1 \Omega \Delta \Delta \Delta \Omega an 7! a1 \Delta \Delta \Delta an; one has H n ((' 1 ) L \Omega n )(!1 \Omega \Delta \Delta \Delta \Omega !n) = (H 1 ' 1 )(!1) \Delta \Delta \Delta (H 1 ' 1 )(!n) in H n ((F\Lambda \Omega ffl X=S ) L \Omega n ). Since H 1 ' 1 = C \Gamma 1 , it follows that H n n (!1 \Omega \Delta \Delta \Delta \Omega !n) = C \Gamma 1 (!1) ^ \Delta \Delta \Delta ^ C \Gamma 1 (!n) in H n F\Lambda \Omega ffl X=S , and therefore that H n n (!1 \Omega \Delta \Delta \Delta \Omega !n) = C \Gamma 1 (!1 ^ \Delta \Delta \Delta ^ !n) = C \Gamma 1 ffi ss(!1 \Omega \Delta \Delta \Delta \Omega !n) by the multiplicative property of the Cartier morphism. Proof of 8.1. Let us choose for each n, a section s of the product ss : (\Omega 1 X=k ) \Omega n ! \Omega n X=k . Still let ss and s be the morphisms relative to X 0 which result from this. Let us choose (cf. 3.9) a lifting Z 0 of X 0 over W2, and define n as in 8.2, with ' 1 = ' 1 Z0 . Let ' n : \Omega n X0=k [\Gamma n] ! F\Lambda \Omega ffl X=S be the composite morphism n ffi s (where s still denotes, by abuse of notation, the corresponding section of (\Omega 1 X=k [\Gamma 1]) \Omega n ! \Omega n X=k [\Gamma n]). It is a question of checking that H n ' n = C \Gamma 1 . However, according to 8.2, if a is a local section of \Omega n X0=k , then H n ' n (a) = (H n n )(sa) = C \Gamma 1 (sssa) = C \Gamma 1 (a); that which completes the proof. Corollary 8.3. Let X be a smooth parallelizable k-scheme, i.e. such that the OX -module \Omega 1 X=k is free (of finite type). Then for X to be DR-decomposable, it is necessary and sufficient that X lifts over W2. 141 8. appendix: parallelizability and ordinary Proof. It suffices to prove the sufficiency. One can assume that k is algebraically closed. Let us choose a rational point x of X over k and an isomorphism ff : \Omega 1 X=k ' f \Lambda E, where f : X ! Spec k is the projection and E is the k-vector space x \Lambda (\Omega 1 X=k ). Via ff, a section of the surjective homomorphism E \Omega n ! \Lambda n E extends to a section of (\Omega 1 X=k ) \Omega n ! \Omega n X=k . Now apply 8.1. Recall (5.5.1) the following definition: Definition 8.4. Let X be a smooth and proper k-scheme. We say that X is ordinary if for any (i; j), one has H j (X; B\Omega i X=k ) = 0, where B\Omega i X=k = d\Omega i\Gamma 1 X=k is the sheaf of boundaries in degree i, of the de Rham complex. This condition is equivalent to H j (X 0 ; F\Lambda B\Omega i X=k ) = 0 for any (i; j). Recall (3.6) that F\Lambda B\Omega i X=k = B i F\Lambda \Omega ffl X=k and F\Lambda Z\Omega i X=k = Z i F\Lambda \Omega ffl X=k are locally free OXmodules of finite type. 8.5. For a given smooth and proper k-scheme X, there exists a link, highlighted by Mehta and Srinivas [Me-Sr], between the properties to be DR-decomposable, parallelizable, and ordinary. This link expresses itself by the means of a concept close to that of DR-decomposability, introduced earlier by Mehta and Ramanathan [Me-Ra], which is the following. We say that a smooth k-scheme X is Frobeniusdecomposable ("Frobenius-split") if the canonical homomorphism OX0 ! F\Lambda OX admits a retraction, i.e. the exact sequence of OX0-modules (cf. 3.5) (8.5.1) 0 ! OX0 ! F\Lambda OX d \Gamma ! F\Lambda B\Omega 1 X=k ! 0 is split. We first observe that if X is Frobenius-decomposable, X is liftable over W2 (or, that which amounts to the same (5.9 (1) (a)), o/!pF\Lambda \Omega ffl X=k is decomposable): The obstruction to the lifting, which is the class of the extension 0 ! OX0 ! F\Lambda OX d \Gamma ! F\Lambda Z\Omega 1 X=k C \Gamma ! \Omega 1 X0=k ! 0; composed with (8.5.1) and the extension (8.5.2) 0 ! F\Lambda B\Omega 1 X=k ! F\Lambda Z\Omega 1 X=k C \Gamma ! \Omega 1 X0=k ! 0; is zero. In general, we are unaware if "Frobenius-decomposable" implies "DRdecomposable". This is the case according to 8.3, if X is parallelizable. But the converse is false. Indeed one has the following result ([Me-Sr] 1.1): If X is a smooth and proper k-scheme, parallelizable, then the following conditions are equivalent: (a) X is Frobenius decomposable; (b) the extension (8.5.2) is split; (c) X is ordinary; (d) (for X of pure dimension d) the homomorphism F \Lambda : H d (X 0 ; OX0) ! H d (X; OX ) induced by the Frobenius is an isomorphism. In particular, if X is ordinary and parallelizable, X lifts over W2 (Nori-Srinivas ([Me-Sr] Appendix) show in fact that for X projective, X lifts to a smooth projective scheme over W ). Moreover - this is the principal result of [Me-Sr] - if k is algebraically closed and X connected, there exists a Galois 'etale lifting Y ! X of order of a power of p such that Y is an abelian variety. If X is projective and smooth over k, ordinary and parallelizable, Nori-Srinivas (loc. cit.) show more precisely that there exists a unique couple (Z; FZ ), where Z luc illusie, frobenius and hodge degeneration 142 is a lifting (projective and smooth) of X over W2 (resp. Wn (n * 2 given), resp. W ) and FZ : Z ! Z 0 a lifting of F : X ! X 0 , where Z 0 is the inverse image of Z by the Frobenius automorphism of W2 (resp. Wn, resp. W ). The existence and uniqueness of this lifting, said canonical, was first established by Serre-Tate [Se-Ta] in the case of abelian varieties. As indicated in 5.5.1, this result admits a converse, without the assumption of parallelizability. Proposition 8.6. Let X be a smooth and proper k-scheme. Assume that there exists schemes Z and Z 0 lifting respectively X and X 0 over W2 and a W2-morphism G : Z ! Z 0 lifting F : X ! X 0 22 . Then X is ordinary. This result was obtained independently by Nakkajima [Na]. 8.7. Proof of 8.6. Let G : Z ! Z 0 be a lifting of F and ' = 'G : L \Omega i X0 [\Gamma i] ! F\Lambda \Omega ffl X the associated homomorphism of complexes, defined in (5.3.1) (one omits =k from the notation of differentials). This homomorphism sends \Omega i X0 into F\Lambda Z\Omega i X (notation of 8.4) and splits the exact sequence (cf. 3.5) (8.7.1) 0 ! F\Lambda B\Omega i X ! F\Lambda Z\Omega i X C \Gamma ! \Omega i X0 ! 0: We prove, by descending induction on i, that H \Lambda (X; B\Omega i X ) = 0 (i.e. that H n (X; B\Omega i X ) = 0 for all n). For i ? dim X; B\Omega i X = 0. Fix i and assume that we proved H \Lambda (X; B\Omega j X ) = 0 for j * i. Then we show that H \Lambda (X; B\Omega i\Gamma 1 X ) = 0. By the exact cohomology sequence associated to the exact sequence (8.7.2) 0 ! F\Lambda Z\Omega i\Gamma 1 X ! F\Lambda \Omega i\Gamma 1 X d \Gamma ! F\Lambda B\Omega i X ! 0; the induction hypothesis implies that for any n, one has H n (X 0 ; F\Lambda Z\Omega i\Gamma 1 X ) , \Gamma ! H n (X; \Omega i\Gamma 1 X ); and therefore (8.7.3) dim H n (X 0 ; F\Lambda Z\Omega i\Gamma 1 X ) = dim H n (X; \Omega i\Gamma 1 X ) = dim H n (X 0 ; \Omega i\Gamma 1 X0 ): The sequence (8.7.1) (relative to i \Gamma 1) being split, implies that the exact sequence of cohomology gives the short exact sequence 0 ! H n (X 0 ; F\Lambda B\Omega i\Gamma 1 X ) ! H n (X 0 ; F\Lambda Z\Omega i\Gamma 1 X ) C \Gamma ! H n (X 0 ; \Omega i\Gamma 1 X0 ) ! 0: The equality (8.7.3) implies that in this situation C is an isomorphism, and therefore that H n (X 0 ; F\Lambda B\Omega i\Gamma 1 X ) = 0, which concludes the proof. Remark 8.8. The reader familiar with logarithmic structures will have observed that 8.6 and its proof extends to the case where k is replaced by a logarithmic point k = (k; M ) with underlying point k and X by a log scheme X = (X; L) log smooth and of Cartier type over k ([Ka2]), proper over k. If one assumes that (X; F ) is lifted over W2(k) (cf. 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