% Analytic Geometry, Chapter VI
% J.P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France

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\titlea{\hfil $L^2$ Hodge Theory and Vanishing Theorems\hfil}{}
\vskip-90pt
\noindent\hskip4cm{\bf Lectures by Jean-Pierre Demailly}
\vskip0pt\noindent
\hskip4cm Universit\'e de Grenoble I, Institut Fourier
\vskip0pt $\phantom{a}$

\titled{Abstract.} These notes are divided in two parts. The goal of
the first part is to give an elementary introduction to Hodge theory
via analytic methods, in the context of Riemannian or K\"ahler
manifolds. The analytic theory rests essentially on the fact that the
De Rham (or Dolbeault) cohomology groups of a compact manifold can be
represented by means of spaces of harmonic forms, once a Riemannian
metric has been chosen. Some basic results about elliptic differential
results are needed, which are most easily treated using
pseudodifferential operators. We investigate in detail the specific
properties of compact K\"ahler manifolds: Hodge decomposition theorem,
hard Lefschetz theorem, description of the Picard group in terms of the
Jacobian variety. Finally, the Hodge-Fr\"olicher spectral sequence is
described, and we use it to derive some general results about Hodge
numbers of compact complex manifolds.

In the second part, we establish the Bochner-Kodaira-Nakano identity
and prove a general version of H\"ormander's $L^2$ estimates for the
$\ovlp$-operator, in the spirit of Andreotti-Vesentini. We then apply
the estimates to derive a few basic vanishing theorems
(Akizuki-Kodaira-Nakano, Kodaira embedding theorem, solution of the
Levi problem). We do this using the recent concept of multiplier ideal
sheaf introduced by A.~Nadel in 1989 (the vanishing theorem with
multiplier ideal sheaves can be seen as a powerful generalization of
the Kawamata-Viehweg vanishing theorem). Finally, by means of a method
introduced by Y.T.~Siu in 1993 (combination of the Riemann-Roch formula
with an argument of noetherian induction for multiplier ideal sheaves),
we obtain reasonably good effective bounds $m_0(n)$ for the very
ampleness of adjoint line bundles $2K_X+mL$, when $L$ is ample and
$m\ge m_0(n)$. These bounds can in turn be used to give an effective
version of the big Matsusaka theorem.
\vskip-20pt$~$

\titlec{}{\phantom{$~$}\newline Contents}
{\bf First Part: $L^2$ Hodge Theory}
   \dotfill p.~2\break
\S 1. Vector Bundles, Connections and Curvature
   \dotfill p.~2\break
\S 2. Differential Operators on Vector Bundles
   \dotfill p.~5\break
\S 3. Basic Results on Elliptic Operators
   \dotfill p.~7\break
\S 4. Hodge Theory of Compact Riemannian Manifolds
   \dotfill p.~11\break
\S 5. Hermitian and K\"ahler Manifolds
   \dotfill p.~16\break
\S 6. Basic Identities of K\"ahler Geometry
   \dotfill p.~19\break
\S 7. Groups $\cH^{p,q}(X,E)$ and Serre Duality
  \dotfill p.~25\break
\S 8. Cohomology of Compact K\"ahler Manifolds
  \dotfill p.~27\break
\S 9. Hodge-Fr\"olicher Spectral Sequence
  \dotfill p.~32\break\phantom{$~$}

\noindent{\bf Second Part: $L^2$ Vanishing Theorems}
  \dotfill p.~36\break
\S 10. Pseudoconvexity and Positivity Concepts
  \dotfill p.~36\break
\S 11. Bochner Technique and Vanishing Theorems
  \dotfill p.~43\break  
\S 12. $L^2$ Estimates and Existence Theorems
  \dotfill p.~46\break
\S 13. Nadel and Kawamata-Viehweg Vanishing Theorems
  \dotfill p.~48\break  
\S 14. Application to the Fujita Conjecture
  \dotfill p.~54\break
References
  \dotfill p.~59

\titleb{\hfil\bgroup\titleafonts First Part: $L^2$ Hodge Theory\egroup\hfil}{}
\titlearunning{First Part: $L^2$ Hodge Theory}
\titleb{\S 1.}{Vector Bundles, Connections and Curvature}
The goal of this section is to recall the most basic definitions of
hermitian differential geometry related to the concepts of connection,
curvature and first Chern class of a line bundle.

\titlec{\S 1.A.}{Dolbeault Cohomology and Sheaf Cohomology}
Let $X$ be a $\bbbc$-analytic manifold of dimension $n$. We denote by
$\Lambda^{p,q}T^\star_X$ the bundle of differential forms of bidegree
$(p,q)$ on $X$, i.e., differential forms which can be written as
$$u=\sum_{|I|=p,\,|J|=q}u_{I,J}dz_I\wedge d\ovl z_J.$$
Here $(z_1\ld z_n)$ denote local holomorphic coordinates,
\hbox{$I=(i_1\ld i_p)$}, \hbox{$J=(j_1\ld j_q)$} are multiindices
(increasing sequences of integers in the range $[1\ld n]$, of lengths
$|I|=p$, $|J|=q$), and we set
$$dz_I:=dz_{i_1}\wedge\ldots\wedge dz_{i_p},\qquad
d\ovl z_J:=d\ovl z_{j_1}\wedge\ldots\wedge d\ovl z_{j_q}.$$
Let $\cA^{p,q}$ be the sheaf of germs of complex valued differential
$(p,q)$-forms with $C^\infty$ coefficients. Recall that the exterior
derivative $d$ splits as $d=d'+d''$ where
$$\eqalign{
d'u&=\sum_{|I|=p,\,|J|=q,1\le k\le n}{\partial u_{I,J}
\over\partial z_k}dz_k\wedge dz_I\wedge d\ovl z_J,\cr
d''u&=\sum_{|I|=p,\,|J|=q,1\le k\le n}{\partial u_{I,J}
\over\partial\ovl z_k}d\ovl z_k\wedge dz_I\wedge d\ovl z_J\cr}$$
are of type $(p+1,q)$, $(p,q+1)$ respectively. The well-known
Dolbeault-Grothendieck lemma asserts that any $d''$-closed
form of type $(p,q)$ with $q>0$ is locally $d''$-exact (this is the
analogue for $d''$ of the usual Poincar\'e lemma for~$d$, see e.g.\
H\"ormander 1966). In other words, the complex of sheaves
$(\cA^{p,\bu},d'')$ is exact in degree $q>0$; in degree $q=0$, $\Ker d''$
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 is a natural $d''$ operator acting
on the space $C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$ of smooth
$(p,q)$-forms with values in~$E$; if $s=\sum_{1\le\lambda\le r}s_\lambda
e_\lambda$ is a $(p,q)$-form expressed in terms of a local holomorphic
frame of $E$, we simply define $d''s:=\sum d''s_\lambda\otimes e_\lambda$,
observing that the holomorphic transition matrices involved in changes
of holomorphic frames do not affect the computation of~$d''$.
It is then clear that the Dolbeault-Grothendieck lemma still
holds for $E$-valued forms. For every integer $p=0,1\ld n$, the
{\it Dolbeault Cohomology} groups $H^{p,q}(X,E)$ are defined to be the
cohomology groups of the complex of global $(p,q)$ forms (graded by $q$):
$$H^{p,q}(X,E)=H^q\big(C^\infty(X,\Lambda^{p,\bu}T^\star_X\otimes E)\big).
\leqno(1.1)$$
Now, let us recall the following fundamental result from sheaf theory
(De Rham-Weil isomorphism theorem): let $(\cL^{\bu},d)$ be a resolution
of a sheaf $\cF$ by acyclic sheaves, i.e.\ a complex of sheaves
$(\cL^\bu,\delta)$ such that there is an exact sequence of sheaves
$$0\lra\cF\buildo j\over\lra\cL^0\buildo \delta^0\over\lra\cL^1\lra\cdots
\lra\cL^q\buildo \delta^q\over\lra\cL^{q+1}\lra\cdots~,$$
and $H^s(X,\cL^q)=0$ for all $q\ge 0$ and $s\ge 1$. Then there
is a fonctorial isomorphism 
$$H^q\big(\Gamma(X,\cL^\bu)\big)\lra H^q(X,\cF).\leqno(1.2)$$
We apply this to the following situation: let $\cA^{p,q}(E)$ be the
sheaf of germs of $C^\infty$ sections of $\Lambda^{p,q}T^\star_X\otimes
E$, Then $(\cA^{p,\bu}(E),d'')$ is a resolution of the locally free
$\cO_X$-module $\Omega^p_X\otimes\cO(E)$ (Dolbeault-Grothendieck
lemma), and the sheaves $\cA^{p,q}(E)$ are acyclic as modules over the
soft sheaf of rings~$\cC^\infty$. Hence by (1.2) we get

\begstat{(1.3) Dolbeault Isomorphism Theorem {\rm (1953)}} For every
holomorphic vector bundle $E$ on~$X$, there is a canonical isomorphism
$$H^{p,q}(X,E)\simeq H^q(X,\Omega^p_X\otimes\cO(E)).\eqno\square$$
\endstat

If $X$ is projective algebraic and $E$ is an algebraic vector bundle,
Serre's GAGA theorem [Ser56] shows that the algebraic sheaf cohomology
group $H^q(X,\Omega^p_X\otimes\cO(E))$ computed with algebraic sections
over Zariski open sets is actually isomorphic to the analytic
cohomology group. 

\titlec{\S 1.B.}{Connections on Differentiable Manifolds}
Let $E$ be a real or complex vector bundle of rank $r$ over a smooth
differentiable manifold~$M$. A {\it connection} $D$ on $E$ is a
linear differential operator of order $1$
$$D:C^\infty(M,\Lambda^qT^\star_M\otimes E) \to C^\infty(M,
\Lambda^{q+1}T^\star_M\otimes E)$$
such that $D$ satisfies the Leibnitz rule
$$D(f\wedge u) = df\wedge u + (-1)^{\deg\,f}f\wedge Du\leqno(1.4)$$
for all forms $f\in C^\infty(M,\Lambda^pT^\star_M)$, $u\in C^\infty(X,
\Lambda^qT^\star_M\otimes E)$. On an open set $\Omega \subset M$ where $E$
admits a trivialization $\tau:E_{|\Omega }\buildo\simeq\over\lra\Omega
\times \bbbc^r$, a connection $D$ can be written
$$Du\simeq_\tau du+\Gamma \wedge u$$
where $\Gamma \in C^\infty(\Omega ,\Lambda^1T^\star_M\otimes\Hom
(\bbbc^r,\bbbc^r))$ is an arbitrary matrix of $1$-forms and $d$ acts
componentwise. It is then easy to check that
$$D^2u\simeq_\tau (d\Gamma +\Gamma \wedge \Gamma )\wedge u\quad
\hbox{on~}\Omega.$$
Since $D^2$ is a globally defined operator, there is a global $2$-form
$$\Theta(D) \in C^\infty(M,\Lambda^2T^\star_M\otimes\Hom(E,E))\leqno(1.5)$$
such that $D^2u=\Theta(D)\wedge u$ for every form $u$ with values in~$E$.

Assume now that $E$ is equipped with a $C^\infty$ euclidean (resp.\ hermitian)
metric along the fibres and that the isomorphism $E_{|\Omega }\simeq\Omega
\times\bbbc^r$ is given by a $C^\infty$ frame $(e_\lambda)$. We then have
a canonical bilinear (resp.\ sesquilinear) pairing
$$\leqalignno{
C^\infty(M,\Lambda^pT^\star_M\otimes E)\times
C^\infty(M,\Lambda^qT^\star_M\otimes E)
&\longrightarrow C^\infty(M,\Lambda^{p+q}T^\star_M\otimes\bbbc)&(1.6)\cr
(u,v) &\longmapsto \{u,v\}&\cr}$$
given by
$$\{u,v\}=\sum_{\lambda ,\mu }u_\lambda \wedge \overline v_\mu \langle
e_\lambda,e_\mu \rangle,\qquad
u=\sum u_\lambda \otimes e_\lambda,\quad 
v=\sum v_\mu \otimes e_\mu.$$
The connection $D$ is said to be {\it hermitian} if it satisfies the
additional property
$$d\{u,v\}=\{Du,v\}+(-1)^{{\rm deg~}u}\{u,Dv\}.$$
Assuming that $(e_\lambda )$ is orthonormal, one easily checks that $D$
is hermitian if and only if $\Gamma^\star=-\Gamma$. In this case
$\Theta(D)^\star=-\Theta(D)$, thus
$$\ii\Theta(D) \in C^\infty(M,\Lambda^2T^\star_M\otimes\Herm(E,E)).$$
\medskip

\begstat{(1.7) Special case} \rm For a complex line bundle $L$ (complex
vector bundle of rank $1$), the connection form $\Gamma$ of a hermitian
connection $D$ can be seen as a $1$-form with purely imaginary coefficients
$\Gamma=\ii A$ ($A$ real). Then we have $\Theta(D)=d\Gamma=\ii dA$.
In particular $\ii\Theta(L)$ is a closed 2-form. The
{\it First Chern class} of $L$ is defined to be the cohomology class
$$c_1(L)_\bbbr=\Big\{{\ii\over 2\pi}\Theta(D)\Big\}\in H^2_\DR(M,\bbbr).$$
The cohomology class is actually independent of the connection, since
any other connection $D_1$ differs by a global 1-form, $D_1u=Du+B\wedge u$,
so that $\Theta(D_1)=\Theta(D)+dB$. It is well-known that $c_1(L)_\bbbr$
is the image in $H^2(M,\bbbr)$ of an integral class $c_1(L)\in
H^2(M,\bbbz)$: let $\cA=\cC^\infty$ be the sheaf of smooth functions on~$M\,$;
by using the exponential exact sequence
$$0\lra\bbbz\lra\cA\buildo e^{2\pi\ii\bu}\over\lra\cA^\star\lra 0,$$
$c_1(L)$ can be defined in \v Cech cohomology as the image of the cocycle
$\{g_{jk}\}\in H^1(M,\cA^\star)$ defining $L$ by the coboundary map
$H^1(M,\cA^\star)\to H^2(M,\bbbz)\,$; see e.g.\ [GH78] for
details.\qed
\endstat

\titlec{\S 1.C.}{Connections on Complex Manifolds}
We now study the specific properties of connections depending
on the complex structure. If $M=X$ is a
complex manifold $X$, every connection $D$ on a complex $\ci$ vector
bundle $E$ can be split in a unique way as a
sum of a $(1,0)$ and of a $(0,1)$-connection, $D=D'+D''$. In a local
trivialization $\tau$ given by a $C^\infty$ frame, one can write 
$$\leqalignno{
D'u &\simeq_\tau d'u + \Gamma '\wedge u,&(1.8')\cr
D''u &\simeq_\tau d''u + \Gamma ''\wedge u,&(1.8'')\cr}$$
with $\Gamma =\Gamma '+\Gamma ''$. The connection is hermitian if and
only if $\Gamma '=-(\Gamma '')^\star$ in any orthonormal frame. Thus there
exists a unique hermitian connection $D$ corresponding to a prescribed
$(0,1)$ part $D''$. 

Assume now that the bundle $E$ itself has a {\it holomorphic} structure.
The unique hermitian connection for which $D''$ is the $d''$ operator
defined in \S~1.A is called the {\it Chern connection} of~$E$. In a local
holomorphic frame $(e_\lambda )$ of $E_{|\Omega }$, the metric is given
by the hermitian matrix $H=(h_{\lambda\mu})$, $h_{\lambda\mu}
=\langle e_\lambda,e_\mu \rangle$. We have
$$\{u,v\}=\sum_{\lambda,\mu}h_{\lambda\mu}u_\lambda \wedge
\ovl v_\mu=u^\dagger\wedge H\ovl v,$$
where $u^\dagger$ is the transposed matrix of $u$, and
easy computations yield
$$\eqalign{d\{u,v\}
&=(du)^\dagger \wedge H\ovl v+(-1)^{\deg u} u^\dagger
\wedge (dH\wedge \ovl v + H\ovl{dv})\cr
&=\big(du +\ovl H^{-1}d'\ovl H \wedge u\big)^\dagger \wedge H\ovl v
+(-1)^{\deg u}u^\dagger\wedge(\ovl{dv+ \ovl H^{-1}d'\ovl H\wedge v)}\cr}$$
using the fact that $dH = d'H+\ovl{d'\ovl H}$ and
$\ovl H^\dagger=H$. Therefore the Chern connection $D$ coincides with the
hermitian connection defined by
$$\left\{\eqalign{
Du&{}\simeq_\tau du+\ovl H^{-1}d'\ovl H \wedge u,\cr
D'&{}\simeq_\tau d'+\ovl H^{-1} d'\ovl H\wedge\bu=
\ovl H^{-1} d'(\ovl H\bu),~~~~D'' = d''.\cr}\right.\leqno(1.9)$$
It is clear from this relations that $D^{\prime 2}=D^{\prime\prime 2}=0$.
Consequently $D^2$ is given by to $D^2=D'D''+D''D'$, and the curvature 
tensor $\Theta(D)$ is of type~$(1,1)$. Since $d'd''+d''d'=0$, we get
$$\eqalign{
(D'D''+D''D')u&\simeq_\tau\ovl H^{-1}d'\ovl H\wedge d''u 
+d''(\ovl H^{-1}d'\ovl H\wedge u)\cr
&=d''(\ovl H^{-1}d'\ovl H)\wedge u.\cr}$$

\begstat{(1.10) Proposition} The Chern curvature tensor $\Theta(E):=
\Theta(D)$ is such that
$$\ii\,\Theta(E) \in C^\infty(X,\Lambda^{1,1}T^\star_X\otimes\Herm(E,E)).$$
If $\tau: E_{\restriction\Omega}\to\Omega\times\bbbc^r$ is a holomorphic
trivialization and if $H$ is the hermitian matrix representing the
metric along the fibres of $E_{\restriction\Omega}$, then
$$\ii\,\Theta(E)\simeq_\tau
\ii\,d''(\ovl H^{-1} d'\ovl H)\quad\hbox{\rm on}~\Omega.\eqno\square$$
\endstat

If $(z_1\ld z_n)$ are holomorphic coordinates on $X$ and 
$(e_\lambda)_{1\le\lambda\le r}$ is an orthonormal frame of $E$, we
can write
$$\ii\Theta(E)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}dz_j\wedge dz_k\otimes e_\lambda^\star\otimes e_\mu.
\leqno(1.11)$$
For any point $\in X$, we say that $(c_{jk\lambda\mu}(x))$ are the
coefficients of the curvature tensor of $E$ at~$x$.

\titleb{\S 2.}{Differential Operators on Vector Bundles}
We first describe some basic concepts concerning differential operators
(symbol, composition, adjunction, ellipticity), in the general setting
of vector bundles. Let $M$ be a $\ci$ differentiable manifold,
$\dim_{\bbbr}M=m$, and let $E$, $F$ be $\bbbk$-vector bundles over $M$,
with $\bbbk=\bbbr$ or $\bbbk=\bbbc$, $\rank E=r$, $\rank F=r'$.

\begstat{(2.1) Definition} A $($linear$)$ differential operator of degree
$\delta$ from $E$ to $F$ is a $\bbbk$-linear operator
$P:C^\infty(M,E)\to C^\infty(M,F)$, $u\mapsto Pu$ of the form
$$Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x),$$
where $E_{\restriction\Omega}\simeq\Omega\times\bbbk^r$,
$F_{\restriction\Omega}\simeq\Omega\times\bbbk^{r'}$ are trivialized
locally on some open chart \hbox{$\Omega\subset M$} equipped with local
coordinates $(x_1\ld x_m)$, and \hbox{$a_\alpha(x)=\big(a_{\alpha\lambda\mu}
(x)\big)_{1\le\lambda\le r',\,1\le\mu\le r}$} are $r'\times r$-matrices
with $\ci$ coefficients on $\Omega$. Here \hbox{$D^\alpha=(\partial/
\partial x_1)^{\alpha_1}\ldots(\partial/\partial x_m)^{\alpha_m}$}
as usual, and \hbox{$u=(u_\mu)_{1\le\mu\le r}$},
\hbox{$D^\alpha u=(D^\alpha u_\mu)_{1\le\mu\le r}$} are viewed as column
matrices.
\endstat

If $t\in\bbbk$ is a parameter and $f\in\ci(M,\bbbk)$, $u\in\ci(M,E)$, a
simple calculation shows that
$e^{-tf(x)}P(e^{tf(x)}u(x))$ is a polynomial of degree $\delta$ in $t$, of
the form
$$e^{-tf(x)}P(e^{tf(x)}u(x))=t^\delta\sigma_P(x,df(x))\cdot u(x)+
\hbox{\rm lower order terms $c_j(x)t^j$, $j<\delta$},$$
where $\sigma_P$ is the homogeneous polynomial map $T^\star_M\to\Hom(E,F)$
defined by
$$T^\star_{M,x}\ni\xi\mapsto\sigma_P(x,\xi)\in\Hom(E_x,F_x),\qquad
\sigma_P(x,\xi)=\sum_{|\alpha|=\delta}a_\alpha(x)\xi^\alpha.\leqno(2.2)$$
Then $\sigma_P(x,\xi)$ is smooth on $T^\star_M$ as a function of $(x,\xi)$,
and is independent of the choice of coordinates or of the trivializations
used for $E$, $F$. We say that $\sigma_P$ is {\it the principal symbol}
of~$P$. The symbol of a composition $Q\circ P$ of differential operators
is simply the product
$$\sigma_{Q\circ P}(x,\xi)=\sigma_{Q}(x,\xi)\sigma_{P}(x,\xi),\leqno(2.3)$$
computed as a product of matrices.

Now, assume that $M$ is oriented and is equipped with a smooth volume
form $dV(x)=\gamma(x)dx_1\wedge\ldots dx_m$, where $\gamma(x)>0$ is a
smooth density. If $E$ is a euclidean or hermitian vector bundle,
we have a Hilbert space $L^2(M,E)$ of global sections $u$ of~$E$ with
measurable coefficients, satisfying the $L^2$ estimate
$$\|u\|^2=\int_M|u(x)|^2\,dV(x)<+\infty.\leqno(2.4)$$
We denote the corresponding $L^2$ inner product by
$$\Ll u,v\Gg=\int_M\langle u(x),v(x)\rangle\,dV(x),\qquad
u,v\in L^2(M,E)\leqno(2.4')$$

\begstat{(2.5) Definition} If $P:\ci(M,E)\to\ci(M,F)$
is a differential operator and both $E$, $F$ are euclidean or hermitian,
there exists a unique differential operator
$$P^\star:\ci(M,F)\to\ci(M,E),$$
called the {\it formal adjoint} of $P$, such that for all sections
$u\in\ci(M,E)$ and $v\in\ci(M,F)$ there is an identity
$$\Ll Pu,v\Gg=\Ll u,P^\star v\Gg,\qquad\hbox{\rm whenever
$\Supp u\cap\Supp v\compact M$}.$$
\endstat

\begproof{} The uniqueness is easy, using the density of smooth compactly
supported forms in $L^2(M,E)$. By a partition of unity argument, it is
enough to show the existence of $P^\star$ locally. Now, let
$Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x)$
be the expansion of $P$ with respect to trivializations of $E$, $F$ given
by orthonormal frames over some coordinate open set $\Omega\subset M$. When
\hbox{$\Supp u\cap\Supp v\compact\Omega$} an integration by parts
yields
$$\eqalign{
\Ll Pu,v\Gg&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}
a_{\alpha\lambda\mu}D^\alpha u_\mu(x)\ovl v_\lambda(x)\,\gamma(x)\,
dx_1\ld dx_m\cr
&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}(-1)^{|\alpha|}
u_\mu(x)\ovl{D^\alpha(\gamma(x)\,\ovl a_{\alpha\lambda\mu}v_\lambda(x)}
\,dx_1\ld dx_m\cr
&=\int_\Omega\langle u,\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}
D^\alpha\big(\gamma(x)\,{}^t\ovl a_\alpha v(x)\big)\rangle\,dV(x).\cr}$$
Hence we see that $P^\star$ exists and is uniquely defined by
$$P^\star v(x)=\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}D^\alpha
\big(\gamma(x)\,{}^t\ovl a_\alpha v(x)\big).\hfill\square\leqno(2.6)$$
\endproof

It follows immediately from (2.6) that the principal symbol of $P^\star$ is
$$\sigma_{P^\star}(x,\xi)=(-1)^\delta\sum_{|\alpha|=\delta}
{}^t\ovl a_\alpha\xi^\alpha=(-1)^\delta\sigma_P(x,\xi)^\star.\leqno(2.7)$$

\begstat{(2.8) Definition} A differential operator $P$ is said to be
elliptic if $\sigma_P(x,\xi)\in\Hom(E_x,F_x)$
is injective for every $x\in M$ and $\xi\in T^\star_{M,x}\ssm\{0\}$.
\endstat

\titleb{\S 3.}{Basic Results on Elliptic Operators}
We assume throughout this section that $M$ is a compact oriented
$m$-dimensional manifold of class $\ci$ equipped with a volume form $dV$,
and we let $E\to M$ be a $\ci$ hermitian vector bundle of rank $r$
over~$M$.

\titlec{\S 3.A.}{Sobolev Spaces}
According to L.~Schwartz, we define the Fr\'echet space $\cS(\bbbr^m)$ of
{\it rapidly decreasing functions} on~$\bbbr^m$ to be the space of
functions $f\in C^\infty(\bbbr^m)$ such that the seminorms
$$p_{k,\ell}(f)=\sup_{x\in\bbbr^m,\,|\alpha|\le k}(1+|x|)^\ell|D^\alpha f(x)|$$
take finite values for all integers $k,\ell\in\bbbn$. For $f\in\cS(\bbbr^m)$
there is a well defined Fourier transform 
$$\wh f(\xi)=\int_{\bbbr^m}f(x)\,e^{-2\pi\ii\,x\cdot\xi}\,d\lambda(x),\qquad
x\cdot\xi=\sum x_j\xi_j.\leqno(3.1)$$
The standard formulas
$$\wh{D^\alpha f}(\xi)=(2\pi\ii\,\xi)^\alpha\wh f(\xi),\qquad
D^\alpha\wh f(\xi)=\big((-2\pi\ii\,x)^\alpha f(x)\big)^\wedge,\qquad
\wh{\wh{\rlap{\hbox{\it f}}\phantom{\rm f}}}(x)=f(-x)\leqno(3.2)$$
show that the Fourier transform is an isomorphism from $\cS(\bbbr^m)$
onto $\cS(\bbbr^m)$. By definition, the space of temperate distributions
is the topological dual space $\cS'(\bbbr^m)$. The Fourier transform
can then be extended as an isomorphism $\cS'(\bbbr^m)\to\cS'(\bbbr^m)$ by the
duality \hbox{formula} $\langle\wh u,f\rangle=\langle u,\wh f\rangle$ for
all $u\in\cS'(\bbbr^m)$, $f\in\cS(\bbbr^m)$ (the duality formula is
indeed an identity if $u,f\in\cS(\bbbr^m)$). Note that the defining formula
(3.1) is still meaningful if $f\in\cE'(\bbbr^m)\subset\cS'(\bbbr^m)$
is a distribution with compact support; in that case $\wh f(\xi)$
extends as an entire holomorphic function on $\bbbc^n$ (the Fourier-Laplace
transform of~$f$).

For any real number $s$, one defines the {\it Sobolev space}
$W^s(\bbbr^m)$ to be the Hilbert space of temperate distributions
$u\in\cS'(\bbbr^m)$ such that the Fourier transform $\wh u$ is a
function in $L^2_\loc$ and
$$\|u\|_s^2=\int_{\bbbr^m}(1+|\xi|^2)^s|\wh u(\xi)|^2d\lambda(\xi)
<+\infty.\leqno(3.3)$$
If $s\in\bbbn$, the first formula (3.2) shows that
$$\|u\|_s^2\sim\int_{\bbbr^m}\sum_{|\alpha|\le s}|D^\alpha u(x)|^2
d\lambda(x),$$
hence $W^s(\bbbr^m)$ is the Hilbert space of functions $u$ such that all
derivatives $D^\alpha u$ or order $|\alpha|\le s$ are in $L^2(\bbbr^m)$.
The dual of $W^s(\bbbr^m)$ can be identified with $W^{-s}(\bbbr^m)$, for
the Plancherel formula implies 
$$\eqalign{\int_{\bbbr^m}u(x)\,\ovl{v(x)}\,d\lambda(x)
&=\int_{\bbbr^m}\wh u(\xi)\,\ovl{\wh v(\xi)}\,d\lambda(\xi)\cr
&=\int_{\bbbr^m}(1+|\xi|^2)^{s/2}\wh u(\xi)\,
(1+|\xi|^2)^{-s/2}\ovl{\wh v(\xi)}\,d\lambda(\xi).\cr}$$

More generally, we denote by $W^s(M,E)$ the Sobolev space of sections
$u:M\to E$ whose components are locally in $W^s(\bbbr^m)$ in each
coordinate chart: we take a finite covering $(\Omega_j)$ of $M$ by coordinate
charts $\Omega_j\simeq\bbbr^m$ on which $E$ is trivial, select an
orthonormal frame $(e_{j,\lambda})_{1\le\lambda\le r}$
of $E_{\restriction\Omega_j}$ in which $u$ is expressed componentwise,
$u=\sum u_{j,\lambda}\,e_{j,\lambda}$, and define
$$\|u\|_s^2=\sum_j\|\psi_j u_j\|_s^2$$
where $(\psi_j)$ is a ``partition of unity'' subordinate to $(\Omega_j)$
such that $\sum\psi_j^2=1$. We claim that $\|~~\|_s$ is independent of all
choices made, up to equivalence of norms. The claim is easy to check once
the continuity of multiplication by test functions and the invariance
under coordinate changes are proved. This is done in the following two
lemmas.

\begstat{(3.4) Lemma} If $\psi\in\cD(\bbbr^m)$, then the map
$u\mapsto\psi u$ extends as a continuous map $W^s(\bbbr^m)\to W^s(\bbbr^m)$.
\endstat

\begproof{} We use the identity
$$\wh{\psi u}(\xi)=\wh\psi\star\wh u(\xi)=
\int_{\bbbr^m}\wh\psi(\xi-\eta)\star\wh u(\eta)\,d\lambda(\eta)$$
in combination with Peetre's inequality
$$(1+|\xi|^2)^s\le 2^{|s|}(1+|\xi-\eta|^2)^{|s|}(1+|\eta|^2)^s.\leqno(3.5)$$
For $s=1$ (resp.\ $s=-1$), Peetre's inequality follows from the obvious
triangle inequality $|\xi|^2\le(|\xi-\eta|+|\eta|)^2\le
2(|\xi-\eta|^2+|\eta|^2)$, resp.\ $|\eta|^2\le 2(|\xi-\eta|^2+|\xi|^2))$),
and for general $s$ it is obtained by raising to the power~$|s|$. This implies
$$(1+|\xi|^2)^{s/2}|\wh{\psi u}(\xi)|\le 2^{|s|/2}
\int_{\bbbr^m}(1+|\xi-\eta|^2)^{|s|/2}|\wh\psi(\xi-\eta)|\,(1+|\eta|^2)^{s/2}
|\wh u(\eta)|\,d\lambda(\eta).$$
The last integral appears to be the convolution of $(1+|\xi|^2)^{|s|/2}
|\wh\psi(\xi)|$ with $(1+|\xi|^2)^{s/2}|\wh u(\xi)|$. We combine this with
Young's inequality $\|f\star g\|_{L^p}\le\|f\|_{L^1}\|g\|_{L^p}$ for $p=2$
to get the desired estimate
$$\|\psi u\|_s\le 2^{|s|/2}\|(1+|\eta|^2)^{|s|/2}\wh\psi(\eta)\|_{L^1}~
\|u\|_s.\eqno{\square}$$
\endproof

\begstat{(3.6) Lemma} If $\varphi:\Omega\to\Omega'$ is a diffeomorphism
of open subsets in $\bbbr^m$, and $u\in W^s(\bbbr^m)$ has its support
contained in a given compact set $K\subset\Omega'$, there is a constant
$C=C(\varphi,K)$ such that $\|u\circ\varphi\|_s\le C\|u\|_s$.
\endstat

\begproof{$($sketch$)$.} By duality, it is enough to consider the case
$s\ge 0\,$; also, since we have
$\|u\|_s\sim\sum_{|\alpha|\le k}\|D^\alpha u\|_{s-k}$ with $k=[s]$,
we may suppose $0<s<1$ (the case $s\in\bbbn$ is anyway clear). However, if
$0<s<1$, the seminorm $\|~~\|_s$ on $W^s(\bbbr^m)$ is equivalent to
$$u\mapsto\Big(\int_{\bbbr^m}|u(x)|^2\,d\lambda(x)+
\int_{\bbbr^m\times\bbbr^m}|u(x)-u(y)|^2|x-y|^{-(m+2s)}\,
d\lambda(x)\,d\lambda(y)\Big)^{1/2}.\leqno(3.7)$$
Taking this for granted, the asserted bound $\|u\circ\varphi\|_s\le C\|u\|_s$
is clear by using a change of variable and the fact that $\varphi$ and
$\varphi^{-1}$ are Lipschitz functions on all compact subsets of
$\Omega$,~$\Omega'$. Now, (3.7) is a direct consequence of the identity
$$\eqalign{
\int_{\bbbr^m\times\bbbr^m}|u(x)-u(y)|^2&|x-y|^{-(m+2s)}\,
d\lambda(x)\,d\lambda(y)\cr
&=\int_{\bbbr^m\times\bbbr^m}|u(x+y)-u(y)|^2|x|^{-(m+2s)}\,
d\lambda(x)\,d\lambda(y)\cr
&=\int_{\bbbr^m\times\bbbr^m}|\wh u(\xi)(e^{2\pi\ii\,x\cdot\xi}-1)|^2
|x|^{-(m+2s)}\,d\lambda(x)\,d\lambda(\xi)\cr
&=c_{m,s}\int_{\bbbr^m}|\xi|^{2s}|\wh u(\xi)|^2\,d\lambda(\xi),\cr}$$
which we get in successive steps by a change of variables $(x,y)\mapsto
(x+y,y)$, a partial Fourier transform in $y$, and finally an application
of the Fubini formula combined with an homogeneity argument.\qed
\endproof

The following basic facts are also needed.

\begstat{(3.8) Sobolev lemma} For every integer $k\in\bbbn$ and every real
number $s>k+{m\over 2}$, one has $W^s(M,E)\subset C^k(M,E)$ and the
inclusion map is continuous.\qed
\endstat

\begproof{} It is enough to show that $W^s(\bbbr^m)\subset C^k(\bbbr^m)$
for $s>k+{m\over 2}$. Suppose first that $k=0$. The Fourier inversion
formula (3.2) yields
$$u(x)=\int_{\bbbr^m}\wh u(\xi)\,e^{2\pi\ii\,x\cdot\xi}d\lambda(\xi).$$
If $s>m/2$, the Cauchy-Schwarz inequality implies
$$\|\wh u\|_{L^1}=\int_{\bbbr^m}|\wh u(\xi)|\,d\lambda(\xi)
\le\Big(\int_{\bbbr^m}(1+|\xi|^2)^{-s}\,d\lambda(\xi)\Big)^{1/2}
\Big(\int_{\bbbr^m}(1+|\xi|^2)^s|\wh u(\xi)|^2\,d\lambda(\xi)\Big)^{1/2}$$
hence $\wh u\in L^1(\bbbr^m)$ and $\|\wh u\|_{L^1}\le C_s\|u\|_s$. We
infer from this that $u$ is continuous and $\|u\|_{L^\infty}\le
C_s\|u\|_s$. If $s>k+{m\over 2}$, we apply this inequality with
$s'=s-k$ and $u$ replaced by $D^\alpha u$ to get
$$\|u\|_{C^k}:=\sup_{|\alpha|\le k}\|D^\alpha u\|_{L^\infty}\le
C_s\sup_{|\alpha|\le k}\|D^\alpha u\|_{s-k}\le C_{s,k}\|u\|_s,$$
as desired.\qed
\endproof

\begstat{(3.9) Corollary} We have
$$C^\infty(M,E)=\bigcap_{s>0}W^s(M,E),\qquad
\cD'(M,E)=\bigcup_{s<0}W^s(M,E).$$
\endstat

\begproof{} The first equality is a direct consequence of the Sobolev lemma.
Now, by dualizing the continuous inclusion $W^s(\bbbr^m)\lhra
C^k(\bbbr^m)$ for $s>k+{m\over 2}$, we find a continuous inclusion
${}^k\cE'(\bbbr^m)\lhra W^{-s}(\bbbr^m)$ of the space of compactly supported
distributions of order $k$ into $W^{-s}(\bbbr^m)$. Thanks to a partition of
unity argument, we get a continuous inclusion ${}^k\cD'(M,E)\lhra
W^{-s}(M,E)$. Now, since $M$ is compact, we have
$\cD'(M,E)=\bigcup{}^k\cD'(M,E)$.\qed
\endproof

\begstat{(3.10) Rellich lemma} For all $t>s$, the inclusion 
$$W^t(M,E)\lhra W^s(M,E)$$ 
is a compact linear operator $($i.e., the image of any bounded subset
of $W^t(M,E)$ is relatively compact in $W^s(M,E))$.
\endstat

\begproof{} By writing any $u\in W^t(M,E)$ as $u=\sum\psi_j u$ with a
partition of unity, we are reduced to proving the following property:
for every compact set $K\subset\bbbr^m$, the injection $W^t_K(\bbbr^m)
\lhra W^s(\bbbr^m)$ of the space of elements of $W^t(\bbbr^m)$ with support
in~$K$ into $W^s(\bbbr^m)$ is a compact operator. Take a sequence
$u_\nu\in W^t_K(\bbbr^m)$ with $\|u_\nu\|_t\le 1$ and a function
$\psi\in\cD(\bbbr^m)$ with $\psi=1$ in a neighborhood of~$K$. Then, as
$D^\alpha\wh u_\nu$ is the Fourier transform of $(-2\pi\ii\,x)^\alpha
u_\nu(x)=\psi(x)(-2\pi\ii\,x)^\alpha u_\nu(x)$, we get for arbitrary
$\alpha\in\bbbn^m$ a uniform bound
$$\int_{\bbbr^m}(1+|\xi|)^{t/2}|D^\alpha\wh u_\nu|^2\,d\lambda(\xi)=
\|\psi(x)(-2\pi\ii\,x)^\alpha u_\nu(x)\|_t^2\le C_\alpha\|u_\nu\|_t^2
\le C_\alpha,$$
thanks to Lemma (3.4). These estimates show that the derivatives
$D^\alpha\wh u_\nu(\xi)$ are uniformly bounded on every compact subset
of~$\bbbr^m$. The Ascoli-Arzela theorem implies that some subsequence
$(\wh u_{\nu_k})$ converges uniformly to a limit on every compact subset
of~$\bbbr^m$. For any $\varepsilon>0$, we fix a radius $R>0$ such that
$R^{s-t}<\varepsilon$. We then find
$$\eqalign{
\|u_{\nu_k}-u_{\nu_\ell}\|_s^2
&=\int_{|\xi|\le R}(1+|\xi|^2)^s|\wh u_{\nu_k}(\xi)-\wh u_{\nu_\ell}(\xi)|^2
\,d\lambda(\xi)+\int_{|\xi|>R}(\ldots)\,d\lambda(\xi)\cr
&\le\Big(\int_{|\xi|\le R}(1+|\xi|^2)^s\,d\lambda(\xi)\Big)
\sup_{\ovl B(0,R)}|\wh u_{\nu_k}-\wh u_{\nu_\ell}|^2
+(1+R^2)^{s-t}\|u_{\nu_k}-u_{\nu_\ell}\|_t^2.\cr}$$
By the inequality $\|u_{\nu_k}-u_{\nu_\ell}\|_t\le
\|u_{\nu_k}\|_t+\|u_{\nu_\ell}\|_t\le 2$ and our choice of $R$ , the
last term is bounded by $4\varepsilon^2$. The uniform convergence of
$(\wh u_{\nu_k})$ on compact sets shows that the term involving the sup
over $\ovl B(0,R)$ is also less than $5\varepsilon^2$ (say) for
$k,\ell$ large enough, hence \hbox{$\|u_{\nu_k}-u_{\nu_\ell}\|_t\le
3\varepsilon$}. Therefore $(u_{\nu_k})$ is a Cauchy sequence in the
Hilbert space $W^s(\bbbr^m)$. We conclude from this that the unit ball of
$W^t_K(\bbbr^m)$ is relatively compact in $W^s(\bbbr^m)$.\qed
\endproof

\titlec{\S 3.B.}{Pseudodifferential Operators}
If $P=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha$ is a differential
operator on $\bbbr^m$, the Fourier inversion formula yields
$$Pu(x)=\int_{\bbbr^m}\sum_{|\alpha|\le\delta}a_\alpha(x)(2\pi\ii\,\xi)^\alpha
\wh u(\xi)\,e^{2\pi\ii\,x\cdot\xi}\,d\lambda(\xi),\qquad
\forall u\in\cD(\bbbr^m),$$
where $\wh u(\xi)=\int_{\bbbr^m}u(x)\,e^{-2\pi\ii\,x\cdot\xi}\,d\lambda(x)$
is the Fourier transform of~$u$. We say that
$$\sigma(x,\xi)=\sum_{|\alpha|\le\delta}a_\alpha(x)(2\pi\ii\,\xi)^\alpha$$
is the {\it total symbol} of $P$.

Given $\sigma$ in a suitable class of functions on $T^\star_{\bbbr^m}$,
the {\it pseudodifferential operator} $\Op_\sigma$ is the operator
defined by
$$\Op_\sigma(u)(x)=\int_{\bbbr^m}\sigma(x,\xi)\,
\wh u(\xi)\,e^{2\pi\ii\,x\cdot\xi}\,d\lambda(\xi),\qquad u\in\cD(\bbbr^m),
\leqno(3.11)$$
Of course, some growth condition of $\sigma(x,\xi)$ with respect to $\xi$
is needed to ensure that the integral is convergent. The
``standard class'' $S^\delta(\bbbr^m)$ of symbols we will consider is
defined as follows: given $\delta\in\bbbr$, $S^\delta(\bbbr^m)$  is the
class of smooth functions $\sigma(x,\xi)$ on $T^\star_{\bbbr^m}$ such that
for every $\alpha,\beta\in\bbbn^m$ and every compact subset $K\subset\bbbr^m$
there is an estimate
$$|D^\alpha_xD_\xi^\beta\sigma(x,\xi)|\le C_{\alpha,\beta}(1+|\xi|)^{
\delta-|\beta|},\qquad\forall(x,\xi)\in K\times\bbbr^m,\leqno(3.12)$$
where $\delta\in\bbbr$ is meant to be the ``degree'' of $\sigma$.
Then $\Op_\sigma(u)$ is a well defined smooth function on $\bbbr^m$, since
$\wh u$ is in the class $\cS(\bbbr^m)$ of rapidly decreasing functions.

For the more general situation of operators over a compact manifold $M$
sending sections of a bundle $E$ into sections of a bundle~$F$, we
introduce corresponding spaces of symbols $S^\delta(M;E,F)$.
The elements of $S^\delta(M;E,F)$ are all functions
$$T^\star_M\ni(x,\xi)\mapsto\sigma(x,\xi)\in\Hom(E_x,F_x)$$
satisfying condition (3.12) in every coordinate chart. We take
a finite covering $(\Omega_j)$ of $M$ by coordinates charts on which
$E,F$ are trivial, and we fix a ``partition of unity'' $(\psi_j)$
subordinate to $\Omega_j$ such that $\sum\psi_j^2=1$. Finally, we define
$$\Op_\sigma(u)=\sum\psi_j\Op_\sigma(\psi_j u),\qquad\forall u\in\ci(M,E),
\leqno(3.13)$$
so as to reduce the calculation to the situation in~$\bbbr^m$.
The basic results of the theory of pseudodifferential operators are
summarized below:

\begstat{(3.14) $W^s$ extension} If $\sigma\in S^\delta(M;E,F)$, then
$\Op_\sigma$ extends in a unique way as a continuous linear operator
$$\Op_\sigma:W^s(M,E)\to W^{s-\delta}(M,F).$$
\endstat

\begproof{} Thanks to Lemma (3.4), it is enough to check that
$\Op_\sigma$ operates continuously from $W^s(\bbbr^m)$ to
$W^{s-\delta}(\bbbr^m)$ whenever $\sigma(x,\xi)$ has compact support
with respect to~$x$, i.e., $\Supp(\sigma)\subset K\times\bbbr^m$ for
some compact set $K\subset\bbbr^m$. In this case, a simple calculation
gives
$$\Op_\sigma(u)^\wedge(\eta)=\int_{\bbbr^m}\wh\sigma_x(\eta-\xi,\xi)\,
\wh u(\xi)\,d\lambda(\xi)$$
where $\wh\sigma_x$ denotes the partial Fourier transform of $\sigma$
in its first argument. In this setting, Condition (3.12) becomes
$$|D_\xi^\beta\wh\sigma_x(\eta,\xi)|\le C_{N,\beta}(1+|\eta|)^{-N}
(1+|\xi|)^{\delta-|\beta|},\qquad\forall(\eta,\xi)\in\bbbr^m\times\bbbr^m,$$
for all $N\in\bbbn$, $\beta\in\bbbn^m$. By the above inequality for
$\beta=0$ and Peetre's inequality for $s-\delta$, we get
$$\eqalign{
(1+|\eta&|^2)^{(s-\delta)/2}|\Op_\sigma(u)^\wedge(\eta)|\le\cr
&\le 2^{|s-\delta|/2}\int_{\bbbr^m}(1+|\eta-\xi|^2)^{|s-\delta|/2}
(1+|\xi|^2)^{(s-\delta)/2}|\wh\sigma_x(\eta-\xi,\xi)|~
|\wh u(\xi)|\,d\lambda(\xi)\cr
&\le C'_N\int_{\bbbr^m}(1+|\eta-\xi|)^{|s-\delta|-N}
(1+|\xi|)^s\,|\wh u(\xi)|\,d\lambda(\xi).\cr}$$
The last integral is the convolution of $(1+|\xi|)^{|s-\delta|-N}$ by
$(1+|\xi|)^s\,|\wh u(\xi)|$. If we take \hbox{$N>|s-\delta|+m$}, then
$(1+|\xi|)^{|s-\delta|-N}$ is in $L^1(\bbbr^n)$, and Young's inequality
implies
$$\|\Op_\sigma(u)\|_{s-\delta}\le C\|u\|_s.\eqno{\square}$$
\endproof

In particular, if $\sigma\in S^{-\infty}(M;E,F):=\bigcap S^\delta(M;E,F)$,
then $\Op_\sigma$ is an operator sending arbitrary distribution
sections of $\cD'(M,E)$ to $\ci(M,F)$ in a continuous way. If
$\sigma\in S^{-\infty}(\bbbr^m)$, the defining formula (3.11) can be rewritten
$$\Op_\sigma(u)(x)=\int_{\bbbr^m}\wh\sigma_\xi(x,y-x)\,u(y)\,d\lambda(y),$$
and by (3.12) we see that $\wh\sigma_\xi(x,y)$ is a smooth function on
$\bbbr^m\times\bbbr^m$ (which is moreover rapidly decreasing in~$y$).
Such an operator is called a smoothing operator. More generally,
one defines the class $\cR$ of {\it smoothing operators} to be
the class of all operators
$$R:\cD'(M,E)\to\ci(M,F),\qquad u\mapsto Ru,\qquad Ru(x)=
\int_M K(x,y)\cdot u(y)\,dV(y),$$
where $K(x,y)\in\Hom(E_y,F_x)$ is a smooth kernel on~$M\times M$.
(NB: a well-known result of L.~Schwartz says in fact that any operator
$R:\cD'(M,E)\to\ci(M,F)$ which is continuous with respect to the
weak topology of $\cD'(M,E)$ and the Fr\'echet topology of $\ci(M,F)$ must
be of this form). When working with pseudodifferential operators, it
is convenient to allow more general operators of the form $\Op_\sigma+R$
where $R\in\cR$ is an arbitrary smoothing operator, and to work only
modulo smoothing operators (according to this philosophy, we will 
deal with symbols only modulo $S^{-\infty}(M;E,F)$). It can be shown,
although we will not do it here, that the class of pseudodifferential
operators $\Op_\sigma+R$ assoicated with all symbols $\sigma\in
S^\delta(M;E,F)$ does not depend on the choice of trivializations,
coordinates or partitions of unity made in (3.13).

\begstat{(3.15) Composition} If $\sigma\in S^\delta(M;E,F)$ and
$\sigma'\in S^{\delta'}(M;F,G)$, there exists a symbol
$\sigma'\sharp\sigma\in S^{\delta+\delta'}(M;E,G)$ such that
$\Op_{\sigma'}\circ\Op_\sigma=\Op_{\sigma'\sharp\sigma}$ mod $\cR$. Moreover
$$\sigma'\sharp\sigma-\sigma'\cdot\sigma\in S^{\delta+\delta'-1}(M;E,G).$$
\endstat

\begstat{(3.16) Definition} A pseudodifferential operator $\Op_\sigma$ of
degree $\delta$ is said to be elliptic if it can be defined by a symbol
$\sigma\in S^\delta(M,E,F)$ such that
$$|\sigma(x,\xi)\cdot u|\ge c|\xi|^\delta|u|,\qquad\forall
(x,\xi)\in T^\star_M,\quad\forall u\in E_x$$
for $|\xi|$ large enough, the estimate being uniform when $x\in M$.
\endstat

If $E$ and $F$ have the same rank, the ellipticity condition implies that
$\sigma(x,\xi)$ is invertible for $\xi$ large. By taking a suitable
cut-off function $\theta(\xi)$ equal to $1$ for $\xi$ large, the function
$\sigma'(x,\xi)=\theta(\xi)\sigma(x,\xi)^{-1}$ defines a symbol in
$S^{-\delta}(M;F,E)$, and by (3.16) we have $\Op_{\sigma'}\circ\Op_\sigma
=\Id+\Op_\rho$, $\rho\in S^{-1}(M;E,E)$. By taking a symbol $\tau$
asymptotically equivalent to the expansion 
$\Id-\rho+\rho^{\sharp 2}+\ldots+(-1)^j\rho^{\sharp j}+\ldots$, we find an
inverse $\Op_{\tau\sharp\sigma'}$ of $\Op_\sigma$ modulo $\cR$.
An easy Corollary is:

\begstat{(3.17) G\aa rding's inequality} Let $P:\ci(M,E)\to\ci(M,F)$ be an
elliptic differential operator of degree $\delta$, where $\rk E=\rk F=r$,
and let $\wt P$ be the extension of 
$P$ to sections with distribution coefficients. For any $u\in W^0(M,E)$
such that $\wt P u\in W^s(M,F)$, then $u\in W^{s+\delta}(M,E)$ and
$$\|u\|_{s+\delta}\le C_s(\|\wt Pu\|_s+\|u\|_0),$$
where $C_s$ is a positive constant depending only on $s$.
\endstat

\begproof{} As $P$ is elliptic, there is a symbol $\sigma\in
S^{-\delta}(M;F,E)$ such that $\Op_\sigma\circ\wt P=\Id+R$, $R\in\cR$.
Then $\|\Op_\sigma(v)\|_{s+\delta}\le C\|v\|_s$ by (3.14), and so
by putting $v=\wt Pu$, we see that $u=\Op_\sigma(\wt P u)-Ru$
satisfies the desired estimate.\qed
\endproof

\titlec{\S 3.C.}{Finiteness Theorem}
We conclude this section by a proof of the following fundamental finiteness
theorem, which is the starting point of $L^2$ Hodge theory.

\begstat{(3.18) Finiteness theorem} Let $E$, $F$ be hermitian vector bundles over $M$
with $\rk E=\rk F=r$, and let $P:C^\infty(M,E)\to C^\infty (M,F)$ be an
elliptic differential operator. Then:
\smallskip
\item{\rm i)} $\Ker P$ is finite dimensional.
\smallskip
\item{\rm ii)} $P\big(\ci(M,E)\big)$ is closed and of finite codimension; 
furthermore, if $P^\star$ is the formal adjoint of $P$, there is a 
decomposition
$$C^\infty(M,F)=P\big(\ci(M,E)\big)\oplus\Ker P^\star$$
as an orthogonal direct sum in $W^0(M,F)=L^2(M,F)$.\smallskip
\endstat

\begproof{} (i) G\aa rding's inequality shows that $\|u\|_{s+\delta}\le C_s\|u\|_0$
for any $u$ in $\Ker P$. Thanks to the Sobolev  lemma, this implies that 
$\Ker P$ is closed in $W^0(M,E)$. Moreover, the unit closed 
$\|~~\|_0$-ball of $\Ker P$ is contained in the $\|~~\|_\delta$-ball of radius
$C_0$, thus compact by the Rellich lemma. Riesz' theorem implies that 
$\dim\Ker P<+\infty$.
\medskip\noindent(ii) We first show that the extension
$$\wt P:W^{s+\delta}(M,E)\to W^s(M,F)$$
has a closed range for any $k$. For every $\varepsilon>0$, there  exists
a finite number of elements $v_1\ld v_N\in W^{s+\delta}(M,F)$,
$N=N(\varepsilon)$,
such that
$$\|u\|_0\le\varepsilon\|u\|_{s+\delta}+\sum^N_{j=1}|\Ll u,v_j\Gg_0|~;
\leqno(3.19)$$
indeed the set
$$K_{(v_j)}=\Big\{ u\in W^{s+\delta}(M,F)~;~\varepsilon\|u\|_{s+\delta}+
\sum^N_{j=1} |\Ll u,v_j\Gg_0|\le 1\Big\}$$
is relatively compact in $W^0(M,F)$ and $\bigcap_{(v_j)}\ovl K_{(v_j)}=\{
0\}$. It follows that there exist elements $(v_j)$ such that 
$\ovl K_{(v_j)}$ is contained in the unit ball of $W^0(M,E)$,~$QED$. 
Substitute $||u||_0$ by the upper bound (3.19) in G\aa rding's inequality;
we get
$$(1-C_s\varepsilon)\|u\|_{s+\delta}\le C_s\Big(\|\wt P u\|_s+\sum^N_{j=1}
|\Ll u,v_j\Gg_0|\Big).$$
Define $G=\big\{u\in W^{s+\delta}(M,E)~;~u\perp v_j,~1\le j\le n\}$ and choose 
$\varepsilon=1/2C_s$.  We obtain
$$\|u\|_{s+\delta}\le 2C_s\|\wt P u\|_s,\qquad\forall u\in G.$$
This implies that $\wt P(G)$ is closed. Therefore
$$\wt P\big(W^{s+\delta}(M,E)\big)=\wt P(G)+{\rm Vect}
\big(\wt P(v_1)\ld\wt P(v_N)\big)$$ 
is closed in $W^s(M,E)$. Take in particular $s=0$. Since $\ci(M,E)$ is dense
in $W^\delta(M,E)$, we see that in $W^0(M,E)$ 
$$\left(\wt P\big(W^\delta(M,E)\big)\right)^\perp=
\Big(P\big(\ci(M,E)\big)\Big)^\perp=\Ker\wt{P^\star}.$$
We have proved that
$$W^0(M,E)=\wt P\big(W^\delta(M,E)\big)\oplus\Ker\wt{P^\star}.
\leqno(3.20)$$
Since $P^\star$ is also elliptic, it follows that $\Ker\wt{P^\star}$
is finite dimensional and that $\Ker\wt{P^\star}=\Ker P^\star$ is
contained in $\ci(M,E)$. Thanks to G\aa rding's inequality, the
decomposition formula (3.20) yields
$$\leqalignno{
W^s(M,E)&=\wt P\big(W^{s+\delta}(M,E)\big)\oplus\Ker P^\star,&(3.21)\cr
\ci(M,E)&=P\big(\ci(M,E)\big)\oplus\Ker P^\star.&(3.22)\cr}$$
\endproof

\titleb{\S 4.}{Hodge Theory of Compact Riemannian Manifolds}
Hodge theory has been invented by W.V.D.~Hodge during the decade
1930-1940 (see [Hod41], [DR55]). The main goal of the theory is to
describe the De Rham cohomology algebra of a Riemannian manifold in
terms of harmonic forms. The main result is that every cohomology class
has a unique harmonic representative.

\titlec{\S 4.A.}{Euclidean Structure of the Exterior Algebra}
Let $(M,g)$ be an oriented Riemannian $\ci$-manifold, $\dim_{\bbbr} M=m$,
and $E\to M$ a hermitian vector bundle of rank $r$ over $M$. We denote 
respectively by $(\xi_1\ld\xi_m)$ and $(e_1\ld e_r)$ orthonormal frames of
$T_M$ and $E$ over an open subset $\Omega\subset M$, and by 
$(\xi^\star_1\ld\xi^\star_m)$,~$(e^\star_1\ld e^\star_r)$ the corresponding 
dual frames of $T^\star_M,~E^\star$. Let $dV$ stand for the Riemannian volume
form on $M$. The exterior algebra $\Lambda^\bu T^\star_M$ has a natural inner 
product $\langle\bu,\bu\rangle$ such that
$$\langle u_1\wedge\ldots\wedge u_p,v_1\wedge\ldots\wedge v_p\rangle=
\det(\langle u_j,v_k\rangle)_{1\le j,k\le p},~~~~u_j,v_k\in T^\star_M
\leqno(4.1)$$
for all $p$, with $\Lambda^\bu T^\star_M=\bigoplus\Lambda^p T^\star_M$ as an
orthogonal sum. Then the family of covectors
\hbox{$\xi^\star_I=\xi^\star_{i_1}\wedge\cdots\wedge\xi^\star_{i_p}$},
$i_1<i_2<\cdots<i_p$, defines an orthonormal basis of $\Lambda^\bu T^\star_M$.
We also denote by $\langle\bu,\bu\rangle$ the corresponding inner product on
$\Lambda^\bu T^\star_M\otimes E$.

\begstat{(4.2) Hodge Star Operator} The Hodge-Poincar\'e-De Rham 
operator $\star$ is the collection of linear maps defined by
$$\star{}:\Lambda^pT^\star_M\to\Lambda^{m-p}T^\star_M,\qquad
u\wedge{}\star v=\langle u,v\rangle\,dV,\qquad
\forall u,v\in\Lambda^pT^\star_M.$$
\endstat

The existence and uniqueness of this operator is easily seen by using
the duality pairing
$$\leqalignno{
\Lambda^pT^\star_M\times\Lambda^{m-p}T^\star_M&{}\lra\bbbr\cr
(u,v)&{}\longmapsto u\wedge v/dV=\sum\varepsilon(I,\complement I)\,
u_Iv_{\complement I},&(4.3)\cr}$$
where $u=\sum_{|I|=p}u_I\,\xi^\star_I$, $v=\sum_{|J|=m-p}v_J\,\xi^\star_J$,
where $\complement I$ stands for the (ordered) complemen\-tary multi-index of
$I$ and $\varepsilon(I,\complement I)$ for the signature of the permutation
\hbox{$(1,2\ld m)\mapsto (I,\complement I)$}. From this, we find
$${}\star v=\sum_{|I|=p}\varepsilon(I,\complement I)v_I\,
\xi^\star_{\complement I}.\leqno(4.4)$$
More generally, the sesquilinear pairing $\{\bu,\bu\}$ defined in
(1.6) yields an operator $\star$ on vector valued forms, such that
$$\leqalignno{
&\star{}:\Lambda^pT^\star_M\otimes E\to\Lambda^{m-p}T^\star_M\otimes E,\qquad	
\{s,{}\star t\}=\langle s,t\rangle\,dV,\qquad
s,t\in\Lambda^pT^\star_M\otimes E,&(4.5)\cr
&\star t=\sum_{|I|=p,\lambda}\varepsilon(I,\complement I)\,
t_{I,\lambda}\,\xi^\star_{\complement I}\otimes e_\lambda&(4.6)\cr}$$
for $t=\sum t_{I,\lambda}\,\xi^\star_I\otimes e_\lambda$. 
Since $\varepsilon(I,\complement I)\varepsilon(\complement I,I)=(-1)^{p(m-p)}=
(-1)^{p(m-1)}$, we get immediately
$$\star\star~t=(-1)^{p(m-1)}t~~~~{\rm on}~~\Lambda^pT^\star_M\otimes E. 
\leqno(4.7)$$
It is clear that $\star$ is an isometry of $\Lambda^\bu T^\star_M\otimes E$. 
We shall also need a variant of the $\star$ operator, namely the 
conjugate-linear operator
$$\#~:~~\Lambda^pT^\star_M\otimes E\lra\Lambda^{m-p}T^\star_M\otimes E^\star$$
defined by $s\wedge\#\,t=\langle s,t\rangle\,dV,$
where the wedge product $\wedge$ is combined with the canonical pairing
\hbox{$E\times E^\star\to\bbbc$}. We have
$$\#\,t=\sum_{|I|=p,\lambda}\varepsilon(I,\complement I)\,\ovl t_{I,\lambda}\,
\xi^\star_{\complement I}\otimes e^\star_\lambda.\leqno(4.8)$$

\begstat{(4.9) Contraction by a Vector Field} Given a tangent vector
$\theta\in T_M$ and a form $u\in\Lambda^pT^\star_M$, the
contraction $\theta\ort u\in\Lambda^{p-1}T^\star_M$ is defined by
$$\theta\ort u\,(\eta_1\ld\eta_{p-1})=u(\theta,\eta_1\ld\eta_{p-1}),~~~~
\eta_j\in T_M.$$
\endstat

In terms of the basis $(\xi_j)$, $\bu\ort\bu$ is the bilinear operation 
characterized by
$$\xi_l\ort(\xi^\star_{i_1}\wedge\ldots\wedge\xi^\star_{i_p})=
\cases{0&if~~$l\notin\{i_1\ld i_p\}$,\cr
(-1)^{k-1}\xi^\star_{i_1}\wedge\ldots\wh{\xi^\star_{i_k}}\ldots\wedge
\xi^\star_{i_p}&if~~$l=i_k$.\cr}$$
This formula is in fact valid even when $(\xi_j)$ is non orthonormal.
A rather easy computation shows that $\theta\ort\bu$ is a {\it derivation}
of the exterior algebra, i.e.\ that
$$\theta\ort(u\wedge v)=(\theta\ort u)\wedge v+(-1)^{{\rm deg}\,u}u\wedge
(\theta\ort v).$$
Moreover, if $\wt\theta=\langle\bu,\theta\rangle\in T^\star_M$, the operator
$\theta\ort\bu$ is the adjoint map of $\wt\theta\wedge\bu$, that is,
$$\langle\theta\ort u,v\rangle=\langle u,\wt\theta\wedge v\rangle,~~~~
\forall u,v\in\Lambda^\bu T^\star_M.\leqno(4.10)$$
Indeed, this property is immediately checked when $\theta=\xi_l$,
$u=\xi^\star_I$, $v=\xi^\star_J$.

\titlec{\S 4.B.}{Laplace-Beltrami Operators}
Let $E$ be a hermitian vector bundle on $M$, and let $D_E$ be a hermitian
connection. We consider the Hilbert space $L^2(M,\Lambda^pT^\star_M\otimes E)$
of $p$-forms $s$ on $M$ with values in $E$, equipped with the $L^2$
inner product
$$\Ll s,t\Gg=\int_M\langle s,t\rangle\,dV$$
already considered in $(2.4')$. Here $\langle s,t\rangle$ is the pointwise
inner product on $\Lambda^pT^\star_M\otimes E$ asso\-cia\-ted with the
Riemannian inner product on $\Lambda^pT^\star_M$ and the hermitian inner
product on~$E$.

\begstat{(4.11) Theorem} The formal adjoint of $D_E$
acting on $C^\infty(M,\Lambda^pT^\star_M\otimes E)$ is given by
$$D_E^\star=(-1)^{mp+1}\star D_E\,\star.$$
\endstat

\begproof{} If $s\in C^\infty(M,\Lambda^pT^\star_M\otimes E)$ and
$t\in C^\infty(M,\Lambda^{p+1}T^\star_M\otimes E)$ have compact 
support, we get
$$\eqalign{
\Ll D_Es,t\Gg&=\int_M\langle D_Es,t\rangle\,dV=\int_M\{D_Es,{}\star t\}\cr
&=\int_M d\{s,{}\star t\}-(-1)^p\{s,D_E\star t\}
=(-1)^{p+1}\int_M\{s,D_E\star t\}\cr}$$
by Stokes' formula. Therefore (4.5) and (4.7) imply
$$\Ll D_Es,t\Gg=(-1)^{p+1}(-1)^{p(m-1)}\int_M
\int_M\{s,\star\star\,D_E\star t\}=(-1)^{mp+1}\Ll s,\star D_E\star t\Gg.$$
The desired formula follows.\qed
\endproof

\begstat{(4.12) Remark} \rm In the case of the trivial connection
$d$ on $E=M\times\bbbc$, the formula becomes $d^\star=(-1)^{m+1}\star
d\star$. If $m$ is even, these formulas reduce to 
$$d^\star=-\star d\,\star,\qquad D_E^\star=-\star D_E\,\star.$$
\endstat

\begstat{(4.13) Definition}
The Laplace-Beltrami operator acting on $\Lambda^pT^\star_M\otimes E$ is
the second order operator
$$\Delta_E=D_ED_E^\star+D_E^\star D_E.$$
In particular, the Laplace-Beltrami operator of $(M,g)$ acting on
$\Lambda^pT^\star_M$ is $\Delta=dd^\star+d^\star d$.
\endstat

It is clear that the Laplace operator $\Delta$ is
formally self-adjoint, i.e.\ $\Ll\Delta_E s,t\Gg=\Ll s,\Delta_E t\Gg$ when
the forms $s,t$ are of class $\ci$ and compactly supported.

\begstat{(4.14) Computation of the symbol} \rm For every smooth function
$f$, Leibnitz's rule yields $e^{-tf}D_E(e^{tf}s)=t\,df\wedge s+D_Es$.
By definition of the symbol, we find
$$\sigma_{D_E}(x,\xi)\cdot s=\xi\wedge s,\qquad \forall\xi)\in T^\star_{M,x},
\quad\forall s\in\Lambda^p T^\star_M\otimes E.$$
By formula (2.7) we have $\sigma_{D_E^\star}=-(\sigma_{D_E})^\star$, hence
$$\sigma_{D_E^\star}(x,\xi)\cdot s=-\wt\xi\ort s$$
where $\wt\xi\in T_M$ is the tangent vector adjoint to $\xi$. From
$\sigma_{\Delta_E}=\sigma_{D_E}\sigma_{D_E^\star}+\sigma_{D_E^\star}
\sigma_{D_E}$, we infer
$$\eqalign{
&\sigma_{\Delta_E}(x,\xi)\cdot s=
-\xi\wedge(\wt\xi\ort s)-\wt\xi\ort(\xi\wedge s)=-(\wt\xi\ort\xi)s,\cr
&\sigma_{\Delta_E}(x,\xi)\cdot s=-|\xi|^2 s.\cr}$$
In particular, $\Delta_E$ is always an elliptic operator. In the special
case where $M$ is an open subset of $\bbbr^m$ equipped with the constant
metric $g=\sum_{i=1}^m dx_i^2$, all operators $d$, $d^\star$, $\Delta$
have constant coefficients. They are completely determined by their
symbol since no other terms of lower order can appear. We then find
easily
$$\eqalign{
s&=\sum_{|I|=p}s_I dx_I,~~~~
ds=\sum_{|I|=p,j} {\partial s_I\over\partial x_j} dx_j\wedge dx_I,\cr
d^\star s&=-\sum_{I,j}{\partial s_I\over\partial x_j} 
{\partial\over\partial x_j}\ort dx_I,\cr
\Delta s&=-\sum_I\Big(\sum_j {\partial^2s_I\over\partial x^2_j}\Big)
dx_I.\cr}$$
Hence $\Delta$ takes the form of the usual elementary Laplace operator,
except for the minus sign.
\endstat

\titlec{\S 4.C.}{Harmonic Forms and Hodge Isomorphism}
Let $E$ be a hermitian vector bundle over a {\it compact} Riemannian manifold
$(M,g)$. We assume that $E$ possesses a {\it flat} hermitian connection $D_E$
(this means that $\Theta(D_E)=D^2_E=0$, or equivalently, that $E$ is given
by a representation $\pi_1(M)\to U(r)$). A fundamental
example is of course the trivial bundle $E=M\times\bbbc$ with the connection
$D_E=d$. Thanks to our flatness assumption, $D_E$ defines a generalized
De Rham complex
$$D_E:C^\infty(M,\Lambda^pT^\star_M\otimes E)\lra
C^\infty(M,\Lambda^{p+1}T^\star_M\otimes E).$$
The cohomology groups of this complex will be denoted by $H^p_{DR}(M,E)$.

The space of {\it harmonic forms of degree $p$} with respect to
the Laplace-Beltrami operator $\Delta_E=D_ED_E^\star+D_E^\star D_E$ is
defined by
$$\cH^p(M,E)=\big\{ s\in C^\infty(M,\Lambda^pT^\star_M\otimes E)~;~
\Delta_E s=0\big\}.\leqno(4.15)$$
As $\Ll\Delta_Es,s\Gg=||D_Es||^2+||D_E^\star s||^2$, we see that 
$s\in\cH^p(M,E)$ if and only if \hbox{$D_Es=D_E^\star s=0$}.

\begstat{(4.16) Theorem} For any $p$, there exists an orthogonal decomposition
$$\eqalign{
&C^\infty(M,\Lambda^pT^\star_M\otimes E)=\cH^p(M,E)\oplus\Im D_E
\oplus\Im D_E^\star,\qquad\hbox{where}\cr
&\Im D_E=D_E\big(C^\infty(M,\Lambda^{p-1}T^\star_M\otimes E)\big),\cr
&\Im D_E^\star=D_E^\star\big(C^\infty(M,\Lambda^{p+1}T^\star_M\otimes E)\big).
\cr}$$
\endstat

\begproof{} It is immediate that $\cH^p(M,E)$ is orthogonal to both
subspaces $\Im D_E$ and $\Im D_E^\star$. The orthogonality of these two
subspaces is also clear, thanks to the assumption $D^2_E=0\,$:
$$\Ll D_Es,D_E^\star t\Gg=\Ll D^2_Es,t\Gg=0.$$
We apply now Th.~3.18 to the elliptic operator $\Delta_E=\Delta^\star_E$
acting on \hbox{$p$-forms,} i.e.\ the operator $\Delta_E:\ci(M,F)\to
\ci(M,F)$ acting on the bundle $F=\Lambda^pT^\star_M\otimes E$. We get
$$\eqalign{
&C^\infty(M,\Lambda^pT^\star_M\otimes E)=\cH^p(M,E)\oplus
\Delta_E\big(\ci(M,\Lambda^pT^\star_M\otimes E)\big),\cr
&\Im\Delta_E=\Im(D_ED_E^\star+D_E^\star D_E)
\subset \Im D_E+\Im D_E^\star.\cr}$$
On the other hand, as $\Im D_E$ and $\Im D_E^\star$ are orthogonal to
$\cH^p(M,E)$, they must be contained in $\Im\Delta_E$.\qed
\endproof

\begstat{(4.17) Hodge isomorphism theorem} The De Rham cohomology group
$H^p_{DR}(M,E)$ is finite dimensional and $H^p_{DR}(M,E)\simeq\cH^p(M,E)$.
\endstat

\begproof{} According to decomposition 4.16, we get
$$\eqalign{
B^p_{DR}(M,E)&=D_E\big(C^\infty(M,\Lambda^{p-1}T^\star_M\otimes E)\big),\cr
Z^p_{DR}(M,E)&=\Ker D_E=(\Im D_E^\star)^\perp=\cH^p(M,E)\oplus\Im D_E.\cr}$$
This shows that every De Rham cohomology class contains a unique harmonic
representative.\qed
\endproof

\begstat{(4.18) Poincar\'e duality} The bilinear pairing
$$H^p_{DR}(M,E)\times H^{m-p}_{DR}(M,E^\star)\lra\bbbc,~~~~
(s,t)\longmapsto\int_M s\wedge t$$
is a non degenerate duality.
\endstat

\begproof{} First note that there exists a naturally defined flat connection
$D_{E^\star}$ such that for any $s\in C^\infty(M,\Lambda^\bu T^\star_M\otimes
E)$, $t\in C^\infty(M,\Lambda^\bu T^\star_M\otimes E^\star)$ we have
$$d(s\wedge t)=(D_Es)\wedge t+(-1)^{\deg s}s\wedge D_{E^\star}t.\leqno(4.19)$$
It is then a consequence of Stokes' formula that the bilinear map
$(s,t)\mapsto\int_M s\wedge t$ can be factorized through cohomology groups.
Let $s\in C^\infty(M,\Lambda^pT^\star_M\otimes E)$. We leave to the reader
the proof of the following formulas (use (4.19) in analogy with the proof of
Th.~4.11):
$$D_{E^\star}(\#\,s)=(-1)^p\#\,D_E^\star s,\quad
\delta_{E^\star}(\#\,s)=(-1)^{p+1}\#\,D_E s,\quad
\Delta_{E^\star}(\#\,s)=\#\,\Delta_E s,\leqno(4.20)$$
Consequently $\#s\in\cH^{m-p}(M,E^\star)$ if and only if $s\in\cH^p(M,E)$. Since
$$\int_M s\wedge\#\,s=\int_M |s|^2\,dV=\|s\|^2,$$
we see that the Poincar\'e pairing has zero kernel in the left hand factor
$\cH^p(M,E)\simeq H^p_{DR}(M,E)$. By symmetry, it has also zero kernel on
the right. The proof is achieved.\qed
\endproof

\titleb{\S 5.}{Hermitian and K\"ahler Manifolds}
Let $X$ be a complex $n$-dimensional manifold. A {\it hermitian metric}
on $X$ is a positive definite hermitian form of class $\ci$ on $T_X\,$;
in a coordinate system $(z_1\ld z_n)$, such a form can be written
\hbox{$h(z)=\sum_{1\le j,k\le n} h_{jk}(z)\,dz_j\otimes d\ovl z_k$,}
where $(h_{jk})$ is a positive hermitian matrix with $\ci$ coefficients.
The {\it fundamental $(1,1)$-form} associated with $h$ is
$$\omega=-\Im h={\ii\over 2}\sum h_{jk} dz_j\wedge d\ovl z_k,~~~~
1\le j,k\le n.$$

\begstat{(5.1) Definition} \smallskip
\item{\rm a)} A hermitian manifold is a pair $(X,\omega)$ where $\omega$
is a $\ci$ positive definite $(1,1)$-form~on~$X$.
\smallskip
\item{\rm b)} The metric $\omega$ is said to be k\"ahler if $d\omega=0$.
\smallskip
\item{\rm c)} $X$ is said to be a K\"ahler manifold if $X$ carries at
least one K\"ahler metric.\smallskip
\endstat

Since $\omega$ is real, the conditions $d\omega=0$, $d'\omega=0$,
$d''\omega=0$ are all equivalent. In local coordinates we see that
$d'\omega=0$ if and only if
$${\partial h_{jk}\over\partial z_l}={\partial h_{lk}\over
\partial z_j},\qquad 1\le j,k,l\le n.$$
A simple computation gives
$${\omega^n\over n!}
=\det(h_{jk})\bigwedge_{1\le j\le n}\Big({\ii\over 2} dz_j\wedge d\ovl z_j\Big)
=\det(h_{jk})\,dx_1\wedge dy_1\wedge\cdots\wedge dx_n\wedge dy_n,$$
where $z_n=x_n+\ii y_n$. Therefore the $(n,n)$-form
$$dV={1\over n!}\omega^n\leqno(5.2)$$
is positive and coincides with the hermitian volume element of $X$. If $X$ is
compact, then $\int_X\omega^n=n!\,\Vol_\omega(X)>0$. This simple remark
already implies that compact K\"ahler manifolds must satisfy some restrictive
topological conditions:

\begstat{(5.3) Consequence} \smallskip
\item{\rm a)} If $(X,\omega)$ is compact K\"ahler and if $\{\omega\}$
denotes the  cohomology class of $\omega$ in $H^2(X,\bbbr)$, then
$\{\omega\}^n\ne 0$.
\smallskip
\item{\rm b)} If $X$ is compact K\"ahler, then $H^{2k}(X,\bbbr)\ne 0$ for
$0\le k\le n$. In fact, $\{\omega\}^k$ is a non zero class in
$H^{2k}(X,\bbbr)$.
\vskip0pt
\endstat

\begstat{(5.4) Example} \rm The complex projective space $\bbbp^n$ is K\"ahler.
A natural K\"ahler metric $\omega$ on $\bbbp^n$, called the {\it Fubini-Study
metric}, is defined by
$$p^\star\omega={\ii\over 2\pi} d'd''\log\big(|\zeta_0|^2+|\zeta_1|^2
+\cdots+|\zeta_n|^2\big)$$
where $\zeta_0,\zeta_1\ld\zeta_n$ are coordinates of $\bbbc^{n+1}$ and
where \hbox{$p:\bbbc^{n+1}\setminus\{0\}\to\bbbp^n$} is the projection.
Let $z=(\zeta_1/\zeta_0\ld\zeta_n/\zeta_0)$ be non homogeneous coordinates
on $\bbbc^n\subset\bbbp^n$. A computation shows that
$$\omega={\ii\over 2\pi}d'd''\log(1+|z|^2)={\ii\over 2\pi}\Theta(\cO(1)),
~~~~\int_{\bbbp^n}\omega^n=1.$$
As the only non vanishing integral cohomology groups of $\bbbp^n$ are
\hbox{$H^{2p}(\bbbp,\bbbz)\simeq\bbbz$} for $0\le p\le n$, we see that
$h=\{\omega\}\in H^2(\bbbp^n,\bbbz)$ is a generator of the
cohomology ring $H^\bu(\bbbp^n,\bbbz)$. In other words,
$H^\bu(\bbbp^n,\bbbz)\simeq\bbbz[h]/(h^{n+1})$ as a ring.
\endstat

\begstat{(5.5) Example} \rm A {\it complex torus} is a quotient
$X=\bbbc^n/\Gamma$ by a lattice $\Gamma$ of rank~$2n$. Then $X$ is a
compact complex manifold. Any positive definite hermitian form
$\omega=\ii\sum h_{jk}dz_j\wedge d\ovl z_k$ with constant coefficients
defines a K\"ahler metric on~$X$.
\endstat

\begstat{(5.6) Example} \rm Every complex submanifold $X$ of a
K\"ahler manifold  $(Y,\omega')$ is K\"ahler with metric
$\omega=\omega'_{\restriction X}$.  Especially, all submanifolds
$X\subset\bbbp^n$ are K\"ahler. In this case, if we take $\omega'$
to be the Fubini-Study metric on $\bbbp^n$, we have the additional
property that $\{\omega\}\in H^2_\DR(X,\bbbr)$ is integral, i.e.,
is the image of an integral class in $H^2(X,\bbbz)$. A K\"ahler
metric $\omega$ with integral cohomology class is said to be a
{\it Hodge metric}.
\endstat

\begstat{(5.7) Example} \rm Consider the complex surface
$$X=(\bbbc^2\setminus\{ 0\})/\Gamma$$
where $\Gamma=\{\lambda^n~;~n\in\bbbz\}$, $\lambda<1$, acts as a
group of homotheties. Since $\bbbc^2\setminus\{ 0\}$ is diffeomorphic
to  $\bbbr^\star_+\times S^3$, we have $X\simeq S^1\times S^3$. 
Therefore $H^2(X,\bbbr)=0$ by K\"unneth's formula, and
property 5.3 b) shows that $X$ is not K\"ahler. More generally,
one can take $\Gamma$ to be an infinite cyclic group generated by
a holomorphic contraction of $\bbbc^2$, of the form
$$\pmatrix{z_1\cr z_2\cr}\longmapsto\pmatrix{\lambda_1z_1\cr\lambda_2z_2\cr},
\qquad\hbox{\rm resp.}\quad\pmatrix{z_1\cr z_2\cr}\longmapsto
\pmatrix{\lambda z_1\cr\lambda z_2+z_1^p\cr},$$
where $\lambda,\lambda_1,\lambda_2$ are complex numbers such that
$0<|\lambda_1|\le|\lambda_2|<1$, \hbox{$0<|\lambda|<1$}, and $p$ a positive
integer. These non K\"ahler surfaces are called {\it Hopf surfaces}.\qed
\endstat

The following Theorem shows that a hermitian metric $\omega$ on $X$
is K\"ahler if and only if the metric $\omega$ is tangent at order $2$ to a 
hermitian metric with constant coefficients at every point of~$X$.

\begstat{(5.8) Theorem} Let $\omega$ be a $\ci$ positive definite $(1,1)$-form
on $X$. In order that $\omega$ be K\"ahler, it is necessary and sufficient
that to every point $x_0\in X$ corresponds a holomorphic coordinate system
$(z_1\ld z_n)$ centered at $x_0$ such that
$$\omega=\ii\sum_{1\le l,m\le n}\omega_{lm}\,dz_l\wedge d\ovl z_m,~~~~
\omega_{lm}=\delta_{lm}+O(|z|^2).\leqno(5.9)$$
If $\omega$ is K\"ahler, the coordinates $(z_j)_{1\le j\le n}$ can be
chosen such that
$$\omega_{lm}=\langle{\partial\over\partial z_l},{\partial\over\partial z_m}
\rangle=\delta_{lm}-\sum_{1\le j,k\le n}c_{jklm}\,z_j\ovl z_k+O(|z|^3)
,\leqno(5.10)$$
where $(c_{jklm})$ are the coefficients of the Chern curvature tensor
$$\Theta(T_X)_{x_0}=\sum_{j,k,l,m} c_{jklm}\,dz_j\wedge d\ovl z_k\otimes
\Big({\partial\over\partial z_l}\Big)^\star
\otimes{\partial\over\partial z_m}\leqno(5.11)$$
associated to $(T_X,\omega)$ at $x_0$. Such a system $(z_j)$
will be called a geodesic coordinate system at $x_0$.
\endstat

\begproof{} It is clear that (5.9) implies $d_{x_0}\omega=0$, so the
condition is sufficient. Assume now that $\omega$ is K\"ahler. Then one can
choose local coordinates $(x_1\ld x_n)$ such that $(dx_1\ld dx_n)$ is an
$\omega$-orthonormal basis of $T_{x_0}^\star X$. Therefore
$$\leqalignno{
\omega&=\ii\sum_{1\le l,m\le n}\wt\omega_{lm}\,dx_l\wedge d\ovl x_m,
~~~~{\rm where}\cr
\qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|)=\delta_{lm}+\sum_{1\le j\le n}
(a_{jlm}x_j+a'_{jlm}\ovl x_j)+O(|x|^2).&(5.12)\cr}$$
Since $\omega$ is real, we have $a'_{jlm}=\ovl a_{jml}\,$; on the other
hand the K\"ahler condition $\partial\omega_{lm}/\partial x_j=
\partial\omega_{jm}/\partial x_l$ at $x_0$ implies $a_{jlm}=a_{ljm}$. Set now
$$z_m=x_m+{1\over 2}\sum_{j,l}a_{jlm}x_jx_l,~~~~1\le m\le n.$$
Then $(z_m)$ is a coordinate system at $x_0$, and
$$\eqalign{
dz_m&=dx_m+\sum_{j,l}a_{jlm}x_jdx_l,\cr
\ii\sum_mdz_m\wedge d\ovl z_m&=\ii\sum_m dx_m\wedge d\ovl x_m
+\ii\sum_{j,l,m}a_{jlm}x_j\,dx_l\wedge d\ovl x_m\cr
&\phantom{=\ii\sum_m dx_m~\wedge d\ovl x_m}
+\ii\sum_{j,l,m}\ovl a_{jlm}\ovl x_j\,dx_m\wedge d\ovl x_l+O(|x|^2)\cr
&=\ii\sum_{l,m}\wt\omega_{lm}\,dx_l\wedge\ovl dx_m+O(|x|^2)=
\omega+O(|z|^2).\cr}$$
Condition (5.9) is proved. Suppose the coordinates $(x_m)$
chosen from the beginning so that (5.9) holds with respect to $(x_m)$.
Then the Taylor expansion (5.12) can be refined into
$$\leqalignno{
\qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|^2))\cr
&=\delta_{lm}+\sum_{j,k}
\big(a_{jklm}x_j\ovl x_k+a'_{jklm}x_jx_k+a''_{jklm}\ovl x_j\ovl x_k\big)+
O(|x|^3).&(5.13)\cr}$$
These new coefficients satisfy the relations
$$a'_{jklm}=a'_{kjlm},~~~~a''_{jklm}=\ovl a'_{jkml},~~~~\ovl a_{jklm}=a_{kjml}.$$
The K\"ahler condition $\partial\omega_{lm}/\partial x_j=\partial
\omega_{jm}/\partial x_l$ at $x=0$ gives the equality
$a'_{jklm}=a'_{lkjm}\,;$ in particular $a'_{jklm}$ is invariant under all
permutations of~$j,k,l$. If we set
$$z_m=x_m+{1\over 3}\sum_{j,k,l}a'_{jklm}\,x_jx_kx_l,~~~~1\le m\le n,$$
then by (5.13) we find
$$\leqalignno{
\qquad dz_m&=dx_m+\sum_{j,k,l}a'_{jklm}\,x_jx_k\,dx_l,~~~~1\le m\le n,\cr
\omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ovl z_m+\ii\sum_{j,k,l,m}a_{jklm}\,x_j
\ovl x_k\,dx_l\wedge d\ovl x_m+O(|x|^3),\cr
\omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ovl z_m+\ii\sum_{j,k,l,m}a_{jklm}\,z_j
\ovl z_k\,dz_l\wedge d\ovl z_m+O(|z|^3).&(5.14)\cr}$$
It is now easy to compute the Chern curvature tensor $\Theta(T_X)_{x_0}$
in terms of the coefficients $a_{jklm}$ and to check that
$c_{jklm}=-a_{jklm}$. This is left as an exercise to the reader.\qed
\endproof

\titleb{\S 6.}{Basic Identities of K\"ahler Geometry}
\titlec{\S 6.A.}{Operators of Hermitian Geometry}
Let $(X,\omega)$ be a hermitian manifold and let $z_j=x_j+\ii y_j$,
$1\le j\le n$, be analytic coordinates at a point $x\in X$ such that
$\omega(x)=\ii\sum dz_j\wedge d\ovl z_j$ is diagonalized at this point.
The associated hermitian form is the $h(x)=2\sum dz_j\otimes d\ovl z_j$
and its real part is the euclidean metric $2\sum (dx_j)^2+(dy_j)^2$. It
follows from this that $|dx_j|=|dy_j|=1/\sqrt{2}$, $|dz_j|=|d\ovl z_j|=1$,
and that $(\partial/\partial z_1\ld\partial/\partial z_n)$ is an orthonormal
basis of $(T^\star_x X,\omega)$. Formula (4.1) with $u_j,v_k$ in the
orthogonal sum $(\bbbc\otimes T_X)^\star=T^\star_X\oplus\ovl{T^\star_X}$
defines a natural inner product on the exterior algebra
$\Lambda^\bu(\bbbc\otimes T_X)^\star$. The norm of a form
$$u=\sum_{I,J}~~u_{I,J} dz_I\wedge d\ovl z_J\in\Lambda^\bu
(\bbbc\otimes T_X)^\star$$
at the given point $x$ is then equal to
$$|u(x)|^2=\sum_{I,J} ~|u_{I,J}(x)|^2.\leqno(6.1)$$

The Hodge $\star $ operator (4.2) can be extended to $\bbbc$-valued forms by 
the formula
$$u\wedge{}\star\ovl v=\langle u,v\rangle\,dV.\leqno(6.2)$$
It follows that $\star $ is a $\bbbc$-linear isometry
$$\star~:~~\Lambda^{p,q} T^\star_X\lra\Lambda^{n-q,n-p} T^\star_X.$$
The usual operators of hermitian geometry are the operators $d,~\delta=
-\star d\,\star ,~\Delta=d\delta+\delta d$ already defined, and their
complex counterparts
$$\left\{\eqalign{
d&=d'+d'',\cr
\delta&=d^{\prime\star}+d^{\prime\prime\star},~~~~d^{\prime\star}=(d')^\star=-\star d''\star,~~~~
d^{\prime\prime\star}=(d'')^\star=-\star d'\star,\cr
\Delta'&=d'd^{\prime\star}+d^{\prime\star}d',~~~~
\Delta''=d''d^{\prime\prime\star}+d^{\prime\prime\star}d''.\cr}\right.\leqno(6.3)$$
Another important operator is the operator $L$ of type (1,1) defined by
$$Lu=\omega\wedge u\leqno(6.4)$$
and its adjoint  $\Lambda=L^\star=\star ^{-1}L\,\star ~:$
$$\langle u,\Lambda v\rangle=\langle Lu,v\rangle.\leqno(6.5)$$

\titlec{\S 6.B.}{Commutation Identities}
If $A,B$ are endomorphisms of the algebra $C^\infty(X,\Lambda^{\bu,\bu}
T^\star_X)$, their graded commutator (or graded Lie bracket) is defined by
$$[A,B]=AB-(-1)^{ab} BA\leqno(6.6)$$
where $a,b$ are the degrees of $A$ and $B$ respectively. If $C$ is another
endomorphism of degree~$c$, the following {\it Jacobi identity} holds
(this is only a formal check):
$$(-1)^{ca}\big[A,[B,C]\big]+(-1)^{ab}\big[B,[C,A]\big]
+(-1)^{bc}\big[C,[A,B]\big]=0.\leqno(6.7)$$
For any $\alpha\in\Lambda^{p,q}T^\star_X$, we still denote by $\alpha$
the endomorphism of type $(p,q)$ on $\Lambda^{\bu,\bu} T^\star_X$ defined by
$u\mapsto\alpha\wedge u$.

Let $\gamma\in\Lambda^{1,1} T^\star_X$ be a real (1,1)-form. There exists an
$\omega$-orthogonal basis $(\zeta_1,\zeta_2\ld\zeta_n)$ in $T_X$ which
diagonalizes both forms $\omega$ and $\gamma\,$:
$$\omega=\ii\sum_{1\le j\le n}\zeta^\star_j\wedge\ovl\zeta^\star_j,~~~~
\gamma=\ii\sum_{1\le j\le n}\gamma_j\,\zeta^\star_j\wedge\ovl\zeta^\star_j,~~
\gamma_j\in\bbbr.$$

\begstat{(6.8) Proposition} For every form $u=\sum u_{J,K}\,\zeta^\star_J 
\wedge\ovl\zeta^\star_K$, one has
$$[\gamma,\Lambda]u=\sum_{J,K}\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}
\gamma_j-\sum_{1\le j\le n}\gamma_j\Big) u_{J,K}\,\zeta^\star_J\wedge\ovl\zeta^\star_K.$$
\endstat

\begproof{} If $u$ is of type $(p,q)$, a brute-force computation yields
$$\eqalignno{
\Lambda u&=\ii(-1)^p\sum_{J,K,l} u_{J,K}\,(\zeta_l\ort\zeta^\star_J)
\wedge(\ovl\zeta_l\ort\ovl\zeta^\star_K),~~~~1\le l\le n,\cr 
\gamma\wedge u&=\ii(-1)^p\sum_{J,K,m}\gamma_m u_{J,K}\,\zeta^\star_m\wedge
\zeta^\star_J\wedge\ovl\zeta^\star_m\wedge\ovl\zeta^\star_K,~~~~1\le m\le n,\cr 
[\gamma,\Lambda] u&=\sum_{J,K,l,m}\gamma_m\,u_{J,K}\Big(
\big(\zeta^\star_l\wedge(\zeta_m\ort\zeta^\star_J)\big)\wedge
\big(\ovl\zeta^\star_l\wedge(\ovl\zeta_m\ort\ovl\zeta^\star_K)\big)\cr
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-\big(\zeta_m\ort(\zeta^\star_l\wedge\zeta^\star_J)\big)\wedge
\big(\ovl\zeta_m\ort(\ovl\zeta^\star_l\wedge\ovl\zeta^\star_K)\big)\Big)\cr
&=\sum_{J,K,m}\gamma_m\,u_{J,K}\Big(
\zeta^\star_m\wedge(\zeta_m\ort\zeta^\star_J)\wedge\ovl\zeta^\star_K\cr
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+\zeta^\star_J\wedge\ovl\zeta^\star_m\wedge(\ovl\zeta_m\ort\ovl\zeta^\star_K)-
\zeta^\star_J\wedge\ovl\zeta^\star_K\Big)\cr
&=\sum_{J,K}\bigg(\sum_{m\in J}\gamma_m+\sum_{m\in K}\gamma_m-\sum_{1\le 
m\le n}\gamma_m\bigg) u_{J,K}\,\zeta^\star_J\wedge\ovl\zeta^\star_K.
&\square\cr}$$
\endproof

\begstat{(6.9) Corollary} For every $u\in\Lambda^{p,q}T^\star_X$, we have
$$[L,\Lambda]u=(p+q-n) u.$$
\endstat

\begproof{} Indeed, if $\gamma=\omega$, we have $\gamma_1=\cdots=\gamma_n=1$.
\qed
\endproof

This result can be generalized as follows: for every
$u\in\Lambda^k(\bbbc\otimes T_X)^\star$, we have
$$[L^r,\Lambda]u=r(k-n+r-1)\,L^{r-1}u.\leqno(6.10)$$
In fact, it is clear that
$$\eqalign{
[L^r,\Lambda]u&=\sum_{0\le m\le r-1}L^{r-1-m}[L,\Lambda]L^m u\cr
&=\sum_{0\le m\le r-1}(2m+k-n)L^{r-1-m}L^m u
=\big(r(r-1)+r(k-n)\big) L^{r-1} u.\cr}$$

We now come to other very important commutation identities. Assume first that
$X=\Omega\subset\bbbc^n$ is an open subset and that
$\omega$ is the standard K\"ahler metric
$$\omega=\ii\sum_{1\le j\le n} dz_j\wedge d\ovl z_j.$$
For any form $u\in C^\infty(\Omega,\Lambda^{p,q}T^\star_X)$ we have
$$\leqalignno{
d'u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
dz_k\wedge dz_I\wedge d\ovl z_J ,&(6.11')\cr
d''u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial\ovl z_k}
d\ovl z_k\wedge dz_I\wedge d\ovl z_J.&(6.11'')\cr}$$
Since the global $L^2$ inner product is given by
$$\Ll u,v\Gg=\int_\Omega\sum_{I,J} u_{I,J}\ovl v_{I,J}\,dV,$$
easy computations analogous to those of Example 4.12 show that
$$\leqalignno{
d^{\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial\ovl z_k}
{\partial\over\partial z_k}\ort (dz_I\wedge d\ovl z_J),&(6.12')\cr
d^{\prime\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
{\partial\over\partial\ovl z_k}\ort (dz_I\wedge d\ovl z_J).&(6.12'')\cr}$$
We first prove a lemma due to Akizuki and Nakano [A-N54].

\begstat{(6.13) Lemma} In $\bbbc^n$, we have $[d^{\prime\prime\star},L]=\ii d'$.
\endstat

\begproof{} Formula (6.1$2''$) can be written more briefly
$$d^{\prime\prime\star}u=-\sum_k {\partial\over\partial\ovl z_k}\ort\Big(
{\partial u\over\partial z_k}\Big).$$
Then we get
$$[d^{\prime\prime\star},L]u=-\sum_k {\partial\over\partial\ovl z_k}\ort\Big(
{\partial\over\partial z_k}(\omega\wedge u)\Big)+\omega\wedge\sum_k 
{\partial\over\partial\ovl z_k}\ort\Big({\partial u\over\partial z_k}\Big).$$
Since $\omega$ has constant coefficients, we have $\displaystyle{\partial\over
\partial z_k}(\omega\wedge u)=\omega\wedge{\partial u\over\partial z_k}$
and therefore
$$\eqalign{
[d^{\prime\prime\star},L]\,u&=-\sum_k\bigg({\partial\over\partial\ovl z_k}\ort
\Big(\omega\wedge {\partial u\over\partial z_k}\Big)-\omega\wedge\Big(
{\partial\over\partial\ovl z_k}\ort{\partial u\over\partial z_k}\Big)\bigg)\cr
&=-\sum_k\Big({\partial\over\partial\ovl z_k}\ort\omega\Big)\wedge
{\partial u\over\partial z_k}.\cr}$$
Clearly $\displaystyle{\partial\over\partial\ovl z_k}\ort\omega=-\ii dz_k$, so
$$[d^{\prime\prime\star},L]\,u=\ii\sum_k dz_k\wedge{\partial u\over\partial z_k}=\ii d'u.
\eqno{\square}$$
\endproof

We are now ready to derive the basic commutation relations in the case of an
arbitrary K\"ahler manifold $(X,\omega)$.

\begstat{(6.14) Theorem} If $(X,\omega)$ is K\"ahler, then
$$\cmalign{&[d^{\prime\prime\star},L]&=&\ii d',~~~~~&[d^{\prime\star},L]&=-&\ii d'',\cr
&[\Lambda,d'']&=-&\ii d^{\prime\star},~~~~~&[\Lambda,d']&=&\ii d^{\prime\prime\star}.\cr}$$
\endstat

\begproof{} It is sufficient to verify the first relation, because the
second one is the conjugate of the first, and the relations of the second line
are the adjoint of those of the first line. If $(z_j)$ is a geodesic
coordinate system at a point $x_0\in X$, then for any $(p,q)$-forms $u,v$
with compact support in a neighborhood of $x_0$, (5.9) implies
$$\Ll u,v\Gg=\int_M\Big(\sum_{I,J}u_{IJ}\ovl v_{IJ}+\sum_{I,J,K,L}a_{IJKL}\,
u_{IJ}\ovl v_{KL}\Big)\,dV,$$
with $a_{IJKL}(z)=O(|z|^2)$ at $x_0$. An integration by parts as in 
(4.12) and (6.1$2''$) yields
$$d^{\prime\prime\star}u=-\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
{\partial\over\partial\ovl z_k}\ort(dz_I\wedge d\ovl z_J)+
\sum_{I,J,K,L}b_{IJKL}\,u_{IJ}\,dz_K\wedge d\ovl z_L,$$
where the coefficients~ $b_{IJKL}$~ are obtained by derivation of the~
$a_{IJKL}$'s. Therefore $b_{IJKL}=O(|z|)$. Since
$\partial\omega/\partial z_k=O(|z|)$, the proof of Lemma~6.13 implies
here  $[d^{\prime\prime\star},L]u=\ii d'u+O(|z|)$, in particular
both terms coincide at every given point $x_0\in X$.\qed
\endproof

\begstat{(6.15) Corollary} If $(X,\omega)$ is K\"ahler, the complex
Laplace-Beltrami operators satisfy 
$$\Delta'=\Delta''={1\over 2}\Delta.$$
\endstat

\begproof{} It will be first shown that $\Delta''=\Delta'$. We have
$$\Delta''=[d'',d^{\prime\prime\star}]=-\ii\big[d'',[\Lambda,d']\big].$$
Since $[d',d'']=0$, Jacobi's identity (6.7) implies 
$$-\big[d'',[\Lambda,d']\big]+\big[d',[d'',\Lambda]\big]=0,$$
hence $\Delta''=\big[d',-\ii[d'',\Lambda]\big]=[d',d^{\prime\star}]=\Delta'$.
On the other hand
$$\Delta=[d'+d'',d^{\prime\star}+d^{\prime\prime\star}]=\Delta'+\Delta''+[d',d^{\prime\prime\star}]
+[d'',d^{\prime\star}].$$
Thus, it is enough to prove:
\endproof

\begstat{(6.16) Lemma} $[d',d^{\prime\prime\star}]=0,~[d'',d^{\prime\star}]=0$.
\endstat

\begproof{} We have $[d',d^{\prime\prime\star}]=-\ii\big[d',[\Lambda,d']\big]$ and 
(6.7) implies
$$-\big[d',[\Lambda,d']\big]+\big[\Lambda,[d',d']\big]+
\big[d',[d',\Lambda]\big]=0,$$
hence $-2\big[d',[\Lambda,d']\big]=0$ and $[d',d^{\prime\prime\star}]=0$. The second
relation 
$[d'',d^{\prime\star}]=0$ is the adjoint of the first.\qed
\endproof

\begstat{(6.17) Theorem} If $(X,\omega)$ is K\"ahler, $\Delta$ commutes with
all operators $\star ,d',d'',d^{\prime\star},d^{\prime\prime\star},L,\Lambda$.
\endstat

\begproof{} The identities $[d',\Delta']=[d^{\prime\star},\Delta']=0$,
$[d'',\Delta'']=[d^{\prime\prime\star},\Delta'']=0$ and $[\Delta,\star]=0$
are immediate. Furthermore, the equality $[d',L]=d'\omega=0$ together
with the Jacobi identity implies
$$[L,\Delta']=\big[L,[d',d^{\prime\star}]\big]=
-\big[d',[d^{\prime\star},L]\big]=\ii[d',d'']=0.$$
By adjunction, we also get $[\Delta',\Lambda]=0$.\qed
\endproof 

\titlec{\S 6.C.}{Primitive Elements and the Hard Lefschetz Theorem}
In this subsection, we prove a fundamental decomposition theorem for the
representation of the unitary group $U(T_X)\simeq U(n)$ acting on
the spaces $\Lambda^{p,q}T^\star_X$ of $(p,q)$-forms. It turns out that
the representation is never irreducible if $0<p,q<n$.

\begstat{(6.18) Definition} A homogeneous element $u\in\Lambda^k
(\bbbc\otimes T_X)^\star$
is called {\it primitive} if $\Lambda u=0$. The space of primitive elements
of total degree $k$ will be denoted
$$\Prim^kT^\star_X=\bigoplus_{p+q=k}\Prim^{p,q}T^\star_X.$$
\endstat

Since the operator $\Lambda$ commutes with the action of $U(n)$ on the
exterior algebra, we conclude that $\Prim^{p,q}T^\star_X\subset
\Lambda^{p,q}T^\star_X$ is a $U(n)$-invariant subspace. One can show that
$\Prim^{p,q}T^\star_X$ is actually an irreducible $U(n)$-representation,
but we will not check this. Now, let
$u\in\Prim^kT^\star_X$. By (6.10) we find
$$\Lambda^s L^r u=\Lambda^{s-1}(\Lambda L^r-L^r\Lambda)u
=r(n-k-r+1)\Lambda^{s-1}L^{r-1}u.$$
By induction, we get for $r\ge s$
$$\Lambda^sL^ru=r(r-1)\cdots (r-s+1)\cdot (n-k-r+1) 
\cdots (n-k-r+s)L^{r-s}u.\leqno(6.19)$$
Apply (6.19) for $r=n+1$. Then $L^{n+1}u$ is of degree $>2n$ and
therefore we have $L^{n+1}u=0$. This gives
$$(n+1)\cdots\big(n+1-(s-1)\big)\cdot (-k)(-k+1)\cdots(-k+s-1)L^{n+1-s}u=0.$$
The integral coefficient is $\ne 0$ if $s\le k$, hence:

\begstat{(6.20) Corollary} If $u\in\Prim^kT^\star_X$, then $L^su=0$ for
$s\ge (n+1-k)_+$.
\endstat

\begstat{(6.21) Corollary} $\Prim^kT^\star_X=0$ for $n+1\le k\le 2n$.
\endstat

\begproof{} Apply Corollary 6.20 with $s=0$.\qed
\endproof

\begstat{(6.22) Primitive decomposition formula} For every
$u\in\Lambda^k(\bbbc\otimes T_X)^\star$, there is a unique decomposition
$$u=\sum_{r\ge (k-n)_+} L^ru_r,~~~~u_r\in\Prim^{k-2r}T^\star_X.$$
Furthermore~ $u_r=\Phi_{k,r} (L,\Lambda)u$~ where $\,\Phi_{k,r}\,$ is a non
commutative polynomial in $L,\Lambda$ with rational coefficients. As
a consequence, there are direct sum decompositions of $U(n)$-representations
$$\eqalign{
&\Lambda^k(\bbbc\otimes T_X)^\star=\bigoplus_{r\ge (k-n)_+}
L^r\Prim^{k-2r}T^\star_X,\cr
&\Lambda^{p,q}T_X^\star=\bigoplus_{r\ge(p+q-n)_+}
L^r\Prim^{p-r,q-r}T^\star_X.\cr}$$
\endstat

\begproof{of the uniqueness of the decomposition.}
Assume that $u=0$ and that $u_r\ne 0$ 
for some~$r$. Let $s$ be the largest integer such that $u_s\ne 0$. Then
$$\Lambda^s u=0=\sum_{(k-n)_+\le r\le s}\Lambda^s L^r u_r=
\sum_{(k-n)_+\le r\le s}\Lambda^{s-r}\Lambda^r L^r u_r.$$
But formula (6.19) shows that $\Lambda^r L^r u_r=c_{k,r} u_r$ for
some non zero integral coefficient $c_{k,r}=r!(n-k+r+1)\cdots (n-k+2r)$.
Since $u_r$ is primitive we get $\Lambda^sL^ru_r=0$ when $r<s$, hence
$u_s=0$, a contradiction.
\endproof

\begproof{of the existence of the decomposition.} We prove by induction on
$s\ge(k-n)_+$ that $\Lambda^s u=0$ implies
$$u=\sum_{(k-n)_+\le r<s} L^ru_r,~~~~
u_r=\Phi_{k,r,s}(L,\Lambda)u\in\Prim^{k-2r}T^\star_X.\leqno(6.23)$$
The Theorem will follow from the step $s=n+1$.

Assume that the
result is true for $s$ and that $\Lambda^{s+1}u=0$. Then $\Lambda^s u$ is
in $\Prim^{k-2s}T^\star_X$. Since $s\ge (k-n)_+$ we have $c_{k,s}\ne 0$
and we set
$$\eqalign{
u_s&={1\over c_{k,s}}\Lambda^s u\in\Prim^{k-2s}T^\star_X,\cr
u'&=u-L^s u_s=\Big(1-{1\over c_{k,s}}L^s\Lambda^s\Big)u.\cr}$$
By formula (6.19), we get
$$\Lambda^s u'=\Lambda^su-\Lambda^s L^s u_s=\Lambda^s u-c_{k,s} u_s=0.$$
The induction hypothesis implies
$$u'=\sum_{(k-n)_+\le r<s} L^ru'_r,~u'_r=\Phi_{k,r,s} (L,\Lambda)
u'\in\Prim^{k-2r}T^\star_X,$$
hence
$u=\sum_{(k-n)_+\le r\le s} L^ru_r$ with
$$\cases{
u_r=u'_r=\Phi_{k,r,s}(L,\Lambda)\Big(1-{\displaystyle 1\over\displaystyle 
c_{k,s}} L^s\Lambda^s\Big)u,~r<s,\cr
u_s={\displaystyle 1\over\displaystyle c_{k,s}}\Lambda^su.\cr}$$
It remains to prove the validity of the decomposition 6.23) for the initial 
step $s=(k-n)_+$, i.e.\ that $\Lambda^su=0$ implies $u=0$. If $k\le n$, then $s=0$ and there is nothing to prove.
We are left with the
case  $k>n$, $\Lambda^{k-n}u=0$. Then $v={}\star u\in\Lambda^{2n-k}
(\bbbc\otimes T_X)^\star$ and $2n-k<n$. Since the decomposition exists in degree
$\le n$ by what we have just proved, we get
$$\eqalign{
v&={}\star u=\sum_{r\ge 0} L^rv_r,~ v_r\in\Prim^{2n-k-2r}T^\star_X,\cr
0&={}\star\Lambda^{k-n}u=L^{k-n}\star  u=\sum_{r\ge 0} L^{r+k-n} v_r,\cr}$$
with degree $(L^{r+k-n}v_r)=2n-k+2(k-n)=k$. The uniqueness part shows that
$v_r=0$ for~all~$r\,$, hence $u=0$. The Theorem is proved.\qed
\endproof

\begstat{(6.24) Corollary} The linear operators
$$\eqalign{
L^{n-k}:\Lambda^k (\bbbc\otimes T_X)^\star
&\lra\Lambda^{2n-k}(\bbbc\otimes T_X)^\star,\cr
L^{n-p-q}:\Lambda^{p,q}T^\star_X&\lra\Lambda^{n-q,n-p}T^\star_X,\cr}$$
are isomorphisms for all integers $k\le n$, $p+q\le n$.
\endstat

\begproof{} For every $u\in\Lambda^k_\bbbc T^\star_X$, the primitive
decomposition $u=\sum_{r\ge 0} L^ru_r$ is mapped bijectively onto
that of $L^{n-k}u\,$:
$$L^{n-k}u=\sum_{r\ge 0} L^{r+n-k}u_r.\eqno{\square}$$
\endproof

\titleb{\S 7.}{Groups $\cH^{p,q}(X,E)$ and Serre Duality}
We now come to the complex aspects of Hodge theory. A major part of
this theory has been developed by K.~Kodaira, S.~Lefschetz and
A.~Weil. The reader will profitably consult the Collected works of
Kodaira and the book of A.~Weil [Wei57]$\,$; see also [Wel80] for a
more recent exposition. 

Let $(X,\omega)$ be a {\it compact hermitian} manifold and $E$ a holomorphic
hermitian vector bundle of rank $r$ over $X$. We denote by  $D_E$ the Chern
connection of $E$, by $D_E^\star=-\star D_E\,\star$ the formal adjoint of 
$D_E$, and by $D^{\prime\star}_E\,,~D^{\prime\prime\star}_E$ the components
of $D_E^\star$ of type $(-1,0)$ and $(0,-1)$. A computation similar to the
one made in 4.14 shows that
$$\sigma_{D''_E}(x,\xi)\cdot s=\xi^{0,1}\wedge s,
\qquad\xi\in{}^\bbbr T^\star_X=\Hom_\bbbr(T_X,\bbbr),\quad s\in E_x,$$
where $\xi^{(0,1)}$ is the $(0,1)$-part of the real $1$-form~$\xi$.
Therefore, we find that the principal part of the operator
$\Delta''_E=D_E''D^{\prime\prime\star}_E+D^{\prime\prime\star}_ED_E''$ is
$$\sigma_{\Delta''_E}(x,\xi)\cdot s=-|\xi^{0,1}|^2 s=-{1\over 2}|\xi|^2 s,$$
and there is of course a similar result for $\Delta'_E$. In particular
$\sigma_{\Delta'_E}=\sigma_{\Delta''_E}={1\over 2}\sigma_{\Delta_E}$
and $\Delta''_E$ is a self-adjoint elliptic operator on each
on each space $C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$. Since
$D^{\prime\prime 2}_E=0$, the following results can be derived in the same
way as those of \S$\,$4.C.

\begstat{(7.1) Theorem} For every bidegree $(p,q)$, there exists an orthogonal 
decomposition
$$C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)=\cH^{p,q}(X,E)\oplus\Im D''_E
\oplus\Im D^{\prime\prime\star}_E$$
where $\cH^{p,q}(X,E)$ is the space of $\Delta''_E$-harmonic forms in
$C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$.
\endstat

The above decomposition shows that the subspace of $d''$-cocycles in
$\ci(X,\Lambda^{p,q}T^\star_X\otimes E)$ is $\cH^{p,q}(X,E)\oplus
\Im D''_E$. From this, we infer

\begstat{(7.2) Hodge isomorphism theorem} The Dolbeault cohomology group
$H^{p,q}(X,E)$ is finite dimensional, and there is an isomorphism
$$H^{p,q}(X,E)\simeq\cH^{p,q}(X,E).\eqno\square$$
\endstat

Another interesting application is a proof of the Serre duality theorem for
compact complex manifolds. See Serre [Ser55] for a proof in a somewhat more
general setting.

\begstat{(7.3) Serre duality theorem} The bilinear pairing
$$H^{p,q}(X,E)\times H^{n-p,n-q}(X,E^\star)\lra\bbbc,\qquad
(s,t)\longmapsto\int_M s\wedge t$$
is a non degenerate duality.
\endstat

\begproof{} Let $s_1\in C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$, $s_2\in C^\infty
(X,\Lambda^{n-p,n-q-1}T^\star_X\otimes E)$.
Since $s_1\wedge s_2$ is of bidegree $(n,n-1)$, we have
$$d(s_1\wedge s_2)=d''(s_1\wedge s_2)=d''s_1\wedge s_2+(-1)^{p+q} s_1
\wedge d''s_2.\leqno(7.4)$$
Stokes' formula implies that the above bilinear pairing can be
factorized through Dolbeault cohomology groups. The $\#$ operator defined
in \S$\,$4.A is such that
$$\#:C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)\lra C^\infty(X,
\Lambda^{n-p,n-q}T^\star_X\otimes E^\star).$$
Furthermore, (4.20) implies
$$d''(\#\,s)=(-1)^{\deg s}\#\,D^{\prime\prime\star}_Es,~~~~
D^{\prime\prime\star}_{E^\star}(\#\,s)=(-1)^{\deg s+1}\#\,D^{\prime\prime\star}_Es,$$
$$\Delta''_{E^\star}(\#\,s)=\#\,\Delta''_Es,$$
where $D_{E^\star}$ is the Chern connection of $E^\star$. Consequently, 
$s\in\cH^{p,q}(X,E)$ if and only if
$\#\,s\in\cH^{n-p,n-q}(X,E^\star)$. Theorem~7.3
is then a consequence of the fact that the integral
$\|s\|^2=\int_X s\wedge\#\,s$ does not vanish unless $s=0$.\qed
\endproof

\titleb{\S 8.}{Cohomology of Compact K\"ahler Manifolds}
\titlec{\S 8.A.}{Bott-Chern Cohomology Groups}
Let $X$ denote for the moment an arbitrary complex manifold. The following
``cohomology groups'' are helpful to describe Hodge theory on
compact complex manifolds which are not necessarily K\"ahler.

\begstat{(8.1) Definition} We define the Bott-Chern cohomology groups
of $X$ to be
$$H_\BC^{p,q}(X,\bbbc)=\big(\ci(X,\Lambda^{p,q}T^\star_X)\cap\Ker d\big)/
d'd''\ci(X,\Lambda^{p-1,q-1}T^\star_X).$$
Then $H_\BC^{\bu,\bu}(X,\bbbc)$ has the structure of a bigraded algebra,
which we call the Bott-Chern cohomology algebra of~$X$.
\endstat

As the group $d'd''\ci(X,\Lambda^{p-1,q-1}T^\star_X)$ is contained in the
Dolbeault as well as in the De Rham coboundary groups
\hbox{$d''\ci(X,\Lambda^{p,q-1}T^\star_X)$},
\hbox{$d\ci(X,\Lambda^{p+q-1}(\bbbc\otimes T_X)^\star)$},
there are cano\-nical morphisms
$$\leqalignno{
&H_\BC^{p,q}(X,\bbbc)\lra H^{p,q}(X,\bbbc),&(8.2)\cr
&H_\BC^{p,q}(X,\bbbc)\lra H^{p+q}_\DR(X,\bbbc),&(8.3)\cr}$$
of the Bott-Chern cohomology to the Dolbeault or De Rham cohomology.
These morphisms are homomorphisms of $\bbbc$-algebras. 
It is also clear from the definition that we have the symmetry
property $H_\BC^{q,p}(X,\bbbc)=\ovl{H_\BC^{p,q}(X,\bbbc)}$.
It can be shown from the Hodge-Fr\"olicher spectral sequence
(see \S$\,$10) that $H_\BC^{p,q}(X,\bbbc)$ is always
finite dimensional if $X$ is compact.

\titlec{\S 8.B.}{Hodge Decomposition Theorem}
We suppose from now on that $(X,\omega)$ is a {\it compact K\"ahler}
manifold. The equality $\Delta=2\Delta''$ shows that $\Delta$ is homogeneous
with respect to bidegree and that there is an orthogonal decomposition
$$\cH^k(X,\bbbc)=\bigoplus_{p+q=k}\cH^{p,q}(X,\bbbc).\leqno(8.4)$$
As $\ovl{\Delta''}=\Delta'=\Delta''$, we also have $\cH^{q,p}(X,\bbbc)=
\ovl{\cH^{p,q}(X,\bbbc)}$. Using the Hodge isomorphism theorems
for the De Rham and Dolbeault cohomology, we get:

\begstat{(8.5) Hodge decomposition theorem} On a compact K\"ahler manifold,
there are cano\-nical isomorphisms
$$\matrix{\displaystyle
H^k(X,\bbbc)\simeq\bigoplus_{p+q=k}H^{p,q}(X,\bbbc)\hfill&\quad
\hbox{$($Hodge decomposition$)$},\hfill\cr
H^{q,p}(X,\bbbc)\simeq\ovl{H^{p,q}(X,\bbbc)}\hfill&\quad
\hbox{$($Hodge symmetry$)$}.\hfill\cr}$$
\endstat

The only point which is not a priori completely clear is that these
isomorphisms are independent of the K\"ahler metric. In order to show that
this is the case, one can use the following Lemma, which allows us to
compare all three types of cohomology groups considered in~\S$\,$8.A.

\begstat{(8.6) Lemma} Let $u$ be a $d$-closed $(p,q)$-form. The following 
properties are equivalent:
\smallskip
\itemitem{\llap{\hbox{${\rm a})$\hfill}}} $u$ is $d$-exact$\,;$
\smallskip
\itemitem{\llap{\hbox{${\rm b'})$\hfill}}} $u$ is $d'$-exact$\,;$
\smallskip
\itemitem{\llap{\hbox{${\rm b''})$\hfill}}} $u$ is $d''$-exact$\,;$
\smallskip
\itemitem{\llap{\hbox{${\rm c})$\hfill}}} $u$ is $d'd''$-exact, i.e.\
$u$ can be written $u=d'd''v$.
\smallskip
\itemitem{\llap{\hbox{${\rm d})$\hfill}}} $u$ is orthogonal to
$\cH^{p,q}(X,\bbbc)$.
\endstat

\begproof{} It is obvious that $c)$ implies a), ${\rm b')}$, ${\rm b'')}$ and
that a) or ${\rm b')}$ or ${\rm b'')}$ implies $d)$. It is thus sufficient
to prove that d) implies~c).  As $du=0$, we have $d'u=d''u=0$, and 
as $u$ is supposed to be orthogonal to $\cH^{p,q}(X,\bbbc)$,
Th.~7.1 implies $u=d''s$, $s\in\ci(X,\Lambda^{p,q-1}T^\star_X)$. By the
analogue of Th.~7.1 for $d'$, we have $s=h+d'v+d^{\prime\star}w$, with
$h\in\cH^{p,q-1}(X,\bbbc)$, $v\in\ci(X,\Lambda^{p-1,q-1}T^\star_X)$ and 
$w\in\ci(X,\Lambda^{p+1,q-1}T^\star_X)$. Therefore 
$$u=d''d'v+d''d^{\prime\star}w=-d'd''v-d^{\prime\star}d''w$$
in view of Lemma~6.16. As $d'u=0$, the component $d^{\prime\star}d''w$
orthogonal to $\Ker d'$ must be zero.\qed
\endproof

From Lemma~8.6 we infer the following Corollary, which in turn implies
that the Hodge decomposition does not depend on the K\"ahler metric.

\begstat{(8.7) Corollary} Let $X$ be a compact K\"ahler manifold. Then
the natural morphisms
$$H^{p,q}_\BC(X,\bbbc)\lra H^{p,q}(X,\bbbc),\qquad
\bigoplus_{p+q=k}H^{p,q}_\BC(X,\bbbc)\lra H^k_\DR(X,\bbbc)$$
are isomorphisms.
\endstat

\begproof{} The surjectivity of $H^{p,q}_\BC(X,\bbbc)\to H^{p,q}(X,\bbbc)$
comes from the fact that every class in $H^{p,q}(X,\bbbc)$ can be represented
by a harmonic $(p,q)$-form, thus by a $d$-closed $(p,q)$-form; the
injectivity means nothing more than the equivalence (8.5$\,{\rm b}'')
\Leftrightarrow(8.5\,$c). Hence $H^{p,q}_\BC(X,\bbbc)\simeq H^{p,q}(X,\bbbc)
\simeq\cH^{p,q}(X,\bbbc)$, and the isomorphism
$\bigoplus_{p+q=k}H^{p,q}_\BC(X,\bbbc)\lra H^k_\DR(X,\bbbc)$
follows from (8.4).\qed
\endproof

Let us quote now two simple applications of Hodge theory. The first of these
is a computation of the Dolbeault cohomology groups of $\bbbp^n$.
As $H^{p,p}(\bbbp^n,\bbbc)\ni\{\omega^p\}\ne 0$ and
\hbox{$H^{2p}_\DR(\bbbp^n,\bbbc)=\bbbc$}, the Hodge decomposition
formula implies:

\begstat{(8.8) Application} The Dolbeault cohomology groups of $\bbbp^n$ are 
$$H^{p,p}(\bbbp^n,\bbbc)=\bbbc\quad\hbox{for}~0\le p\le n,\qquad
H^{p,q}(\bbbp^n,\bbbc)=0\quad\hbox{for}~p\ne q.\eqno\square$$
\endstat

\begstat{(8.9) Proposition} Every holomorphic $p$-form on a
compact K\"ahler mani\-fold $X$ is $d$-closed.
\endstat

\begproof{} If $u$ is a holomorphic form of type $(p,0)$ then
$d''u=0$. Furthermore $d^{\prime\prime\star}u$ is of type $(p,-1)$, hence
$d^{\prime\prime\star}u=0$. Therefore $\Delta u=2\Delta''u=0$, which implies 
$du=0$.\qed
\endproof

\begstat{(8.10) Example} \rm Consider the {\it Heisenberg group}
$G\subset{\rm Gl}_3(\bbbc)$, defined as the subgroup of matrices
$$M=\pmatrix{
1~~&\hfill x~~&\hfill z\hfill\cr
0~~&\hfill 1~~&\hfill y\hfill\cr
0~~&\hfill 0~~&\hfill 1\hfill\cr}\quad,\quad (x,y,z)\in\bbbc^3.$$
Let $\Gamma$ be the discrete subgroup of matrices with entries 
$x,y,z\in\bbbz[\ii]$ (or more generally in the ring of integers of an
imaginary quadratic field). Then $X=G/\Gamma$ is a compact complex
3-fold, known as the {\it Iwasawa manifold}. The equality
$$M^{-1}dM=\pmatrix{0&dx&dz-xdy\cr 0&0&dy\cr 0&0&0\cr}$$
shows that $dx,~dy,~dz-xdy$ are left invariant holomorphic $1$-forms on
$G$. These forms induce holomorphic $1$-forms on the quotient $X=G/\Gamma$.
Since $dz-xdy$ is not $d$-closed, we see that $X$ cannot be K\"ahler.
\endstat

\titlec{\S 8.C.}{Primitive Decomposition and Hard Lefschetz Theorem}
We first introduce some standard notation. The {\it Betti numbers} and
{\it Hodge numbers} of $X$ are by definition
$$b_k=\dim_\bbbc H^k(X,\bbbc),~~~~h^{p,q}=\dim_\bbbc H^{p,q}(X,\bbbc).
\leqno(8.11)$$
Thanks to Hodge decomposition, these numbers satisfy the relations
$$b_k=\sum_{p+q=k} h^{p,q},~~~~h^{q,p}=h^{p,q}.\leqno(8.12)$$
As a consequence, the Betti numbers $b_{2k+1}$ of a compact K\"ahler manifold
are even. Note that the Serre duality theorem gives the additional
relation \hbox{$h^{p,q}=h^{n-p,n-q}$}, which holds as soon as $X$ is
compact. The existence of primitive decomposition implies other interesting
specific features of the cohomology algebra of compact K\"ahler manifolds.

\begstat{(8.13) Lemma} If $u=\sum_{r\ge (k-n)_+}L^ru_r$ is the primitive
decomposition of a harmonic $k$-form $u$, then all components
$u_r$ are harmonic.
\endstat

\begproof{} Since  $[\Delta,L]=0$, we get $0=\Delta u=\sum_r L^r\Delta
u_r$, hence $\Delta u_r=0$ by uniqueness.\qed
\endproof

Let us denote by $\cH^k_\prim(X,\bbbc)=\bigoplus_{p+q=k}\cH^{p,q}_\prim
(X,\bbbc)$ the space of primitive harmonic $k$-forms and let $h_\prim^{p,q}$
be  the dimension of the $(p,q)$-component. Lemma 8.13 yields
$$\leqalignno{
\cH^{p,q}(X,\bbbc)&=\bigoplus_{r\ge (p+q-n)_+}L^r\cH_\prim^{p-r,q-r}(X,\bbbc),
&(8.14)\cr
h^{p,q}&=\sum_{r\ge (p+q-n)_+}h_\prim^{p-r,q-r}.&(8.15)\cr}$$
Formula (8.15) can be rewritten
$$\cases{\phantom{\Big(}
{\rm if}~~p+q\le n,~~h^{p,q}=h_\prim^{p,q}+h_\prim^{p-1,q-1}+\cdots\cr
\phantom{\Big(}
{\rm if}~~p+q\ge n,~~h^{p,q}=h_\prim^{n-q,n-p}+h_\prim^{n-q-1,n-p-1}+\cdots.
\cr}\leqno(8.15')$$

\begstat{(8.16) Corollary} The Hodge and Betti numbers satisfy the inequalities
\smallskip
\item{\rm a)} if $k=p+q\le n$,
then~ $h^{p,q}\ge h^{p-1,q-1},~~b_k\ge b_{k-2}$,
\smallskip
\item{\rm b)} if $k=p+q\ge n$,
then~ $h^{p,q}\ge h^{p+1,q+1},~~b_k\ge b_{k+2}$.\qed
\endstat

Another important result of Hodge theory (which is in fact a direct
consequence of Cor.~6.24) is the

\begstat{(8.17) Hard Lefschetz theorem} The mappings
$$\cmalign{
&L^{n-k}~:~~\hfill H^k(X,\bbbc)&\lra H^{2n-k}(X,\bbbc),~~~~&k\le n,\cr
&L^{n-p-q}~:~~H^{p,q}(X,\bbbc)&\lra H^{n-q,n-p}(X,\bbbc),~~~~&p+q\le n,\cr}$$
are isomorphisms.\qed
\endstat

Another way of stating the Hard Lefschetz theorem is to introduce the
{\it Hodge-Riemann bilinear form} on $H^k_\DR(X,\bbbc)$ defined by
$$Q(u,v)=(-1)^{k(k-1)/2}\int_X u\wedge v\wedge\omega^{n-k}.\leqno(8.18)$$
The Hard Lefschetz theorem combined with Poincar\'e duality tells us that
$Q$ is non degenerate. Moreover $Q$ is of parity $(-1)^k$ (symmetric if
$k$ is even, alternate if $k$ is odd). When $\omega$ is a {\it Hodge
metric}, that is, a K\"ahler metric such that $\{\omega\}\in H^2(X,\bbbz)$,
it is clear that $Q$ takes integral values on the lattice
$H^k(X,\bbbz)/({\rm torsion})$. The Hodge-Riemann bilinear form satisfies
the following additional properties: for $p+q=k$,
$$\leqalignno{
&Q(H^{p,q},H^{p',q'})=0\qquad\hbox{\rm unless $(p',q')=(q,p)$},&(8.19')\cr
&\hbox{If $0\ne u\in H^{p,q}_\prim(X,\bbbc)$, then
$\ii^{p-q}Q(u,\ovl u)=\|u\|^2>0$.}&(8.19'')\cr}$$
In fact $(8.19')$ is clear, and $(8.19'')$ will follow if we show that
every primitive $(p,q)$-form $u$ satisfies
$$(-1)^{k(k-1)/2}\ii^{p-q}\omega^{n-k}\wedge\ovl u=\star\ovl u.$$
Since $\Prim^{p,q}T^\star_X$ is an irreducible $U(n)$-representation, it
is enough to check the formula for a single well chosen $(p,q)$-form $u$.
We simply take $u=dz_1\wedge\ldots\wedge dz_p\wedge d\ovl z_{p+1}\wedge
\ldots\wedge dz_{p+q}$ in an orthormal basis for $\omega$. The actual
computation is left as an exercise for the reader.

\titlec{\S 8.D.}{Description of the Picard Group}
Another important application of Hodge theory is a description of the Picard 
group $H^1(X,\cO^\star)$ of a compact K\"ahler manifold. We assume here that
$X$ is connected. The exponential exact
sequence $0\to\bbbz\to\cO\to\cO^\star\to 1$ gives
$$0\lra H^1(X,\bbbz)\lra H^1(X,\cO)\lra H^1(X,\cO^\star)
\buildo c_1\over\lra H^2(X,\bbbz)\lra H^2(X,\cO)\eqno(8.20)$$
because the map $\exp(2\pi i\bu):H^0(X,\cO)=\bbbc\lra H^0(X,\cO^\star)=
\bbbc^\star$ is onto. 
We have $H^1(X,\cO)\simeq H^{0,1}(X,\bbbc)$ by the Dolbeault isomorphism.
The dimension of this group is called the {\it irregularity of $X$} and
is usually denoted
$$q=q(X)=h^{0,1}=h^{1,0}.\leqno(8.21)$$
Therefore we have $b_1=2q$ and
$$H^1(X,\cO)\simeq\bbbc^q,~~~~H^0(X,\Omega^1_X)=H^{1,0}(X,\bbbc)
\simeq\bbbc^q.\leqno(8.22)$$

\begstat{(8.23) Lemma} The image of $H^1(X,\bbbz)$ in $H^1(X,\cO)$ is a lattice.
\endstat

\begproof{} Consider the morphisms
$$H^1(X,\bbbz)\lra H^1(X,\bbbr)\lra H^1(X,\bbbc)\lra H^1(X,\cO)$$
induced by the inclusions $\bbbz\subset\bbbr\subset\bbbc\subset\cO$. Since the
\v Cech cohomology groups with values in $\bbbz$, $\bbbr$ can be computed
by finite acyclic coverings, we see that $H^1(X,\bbbz)$ is a finitely 
generated $\bbbz$-module and that the image of $H^1(X,\bbbz)$ in $H^1(X,\bbbr)$
is a lattice. It is enough to check that the map $H^1(X,\bbbr)\lra H^1(X,\cO)$
is an isomorphism. However, the commutative \hbox{diagram}
$$\cmalign{
&0&\lra&\bbbc&\lra&~\cA^0&\buildo d\over\lra&~\cA^1&\buildo d\over\lra&~\cA^2
&\lra&\cdots\cr
&&&\big\downarrow&&~\big\downarrow&&~\big\downarrow&&~\big\downarrow&&\cr
&0&\lra&\cO&\lra&\cA^{0,0}&\buildo d''\over\lra&\cA^{0,1}&\buildo d''\over\lra&
\cA^{0,2}&\lra&\cdots\cr}$$
shows that the map $H^1(X,\bbbr)\lra H^1(X,\cO)$ corresponds in De Rham and
Dolbeault cohomologies to the composite mapping
$$H^1_{DR}(X,\bbbr)\subset H^1_{DR}(X,\bbbc)\lra H^{0,1}(X,\bbbc).$$
Since~ $H^{1,0}(X,\bbbc)$ and~ $H^{0,1}(X,\bbbc)$ are complex conjugate
subspaces in
$H^1_{DR}(X,\bbbc)$, we conclude that $H^1_{DR}(X,\bbbr)\lra H^{0,1}(X,\bbbc)$
is an isomorphism.\qed
\endproof

As a consequence of this lemma, $H^1(X,\bbbz)\simeq\bbbz^{2q}$.
The $q$-dimensional complex torus
$$\Jac(X)=H^1(X,\cO)/H^1(X,\bbbz)\leqno(8.24)$$
is called the {\it Jacobian variety of $X$} and is isomorphic to the
subgroup of $H^1(X,\cO^\star)$ corresponding to line bundles of zero
first Chern class. On the other hand, the kernel of
$$H^2(X,\bbbz)\lra H^2(X,\cO)=H^{0,2}(X,\bbbc)$$
which consists of integral cohomology classes of type~$(1,1)$,
is equal to the image of $c_1$ in $H^2(X,\bbbz)$. This subgroup is called
the {\it Neron-Severi group} of~$X$, and is denoted $NS(X)$. The exact 
sequence (8.20) yields
$$0\lra\Jac(X)\lra H^1(X,\cO^\star)\buildo c_1\over\lra NS(X)\lra 0.\leqno(8.24)$$
The Picard group $H^1(X,\cO^\star)$ is thus an extension of the complex torus
$\Jac(X)$ by the finitely generated $\bbbz$-module $NS(X)$.

\begstat{(8.25) Corollary} The Picard group of $\bbbp^n$ is $H^1(\bbbp^n,\cO^\star)
\simeq\bbbz$, and every line bundle over $\bbbp^n$ is isomorphic to one of the 
line bundles $\cO(k)$, $k\in\bbbz$.
\endstat

\begproof{} We have $H^k(\bbbp^n,\cO)=H^{0,k}(\bbbp^n,\bbbc)=0$ for
$k\ge 1$ by Appl.~8.8, thus $\Jac({\bbbp^n})=0$ and
$NS(\bbbp^n)=H^2(\bbbp^n,\bbbz)\simeq\bbbz$. Moreover,
$c_1\big(\cO(1)\big)$ is a generator of $H^2(\bbbp^n,\bbbz)$.\qed
\endproof

\titleb{\S 9.}{Hodge-Fr\"olicher Spectral Sequence}
Let $X$ be a {\it compact} complex $n$-dimensional manifold. We
consider the double complex $K^{p,q}=\ci(X,\Lambda^{p,q}T^\star_X)$
with its total differential $d=d'+d''$. The {\it Hodge-Fr\"olicher
spectral sequence} is by definition the spectral sequence associated
with this double complex.

Let us first recall the algebraic machinery of spectral sequences, for
an arbitrary double complex $(K^{p,q},d'+d'')$ of modules over some ring.
We assume here for simplicity that \hbox{$K^{p,q}=0$} if $p<0$ or
$q<0$. First, one associates with $K^{\bu,\bu}$ the simple complex
\hbox{$K^l=\bigoplus_{p+q=l}K^{p,q}$} equipped with the total differential
$d=d'+d''$. Then $K^\bu$ has a decreasing filtration by subcomplexes
$F^pK^\bu$ such that
$$F^pK^l=\bigoplus_{p\le j\le l}K^{j,l-j}.\leqno(9.1)$$
One obtains an induced decreasing filtration on the total cohomology groups
$H^l(K^\bu)$ by putting
$$F^pH^l(K^\bu):=\Im\big(H^l(F^pK^\bu)\to H^l(K^\bu)\big).\leqno(9.2)$$
Now, the theory of spectral sequences (see e.g. [God57]) tells us that there
is a sequence of double complexes $E_r^{\bu,\bu}$, $r\ge 1$, equipped with
differentials $d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}$ of bidegree $(r,-r+1)$, such
that $E_{r+1}=H^\bu(E_r)$ is computed inductively as the cohomology of the
complex $(E_r^{\bu,\bu},d_r)$, and that the limit
$$E_\infty^{p,q}=\lim_{r\to+\infty}E_r^{p,q}=F^pH^l(K^\bu)/F^{p+1}H^l(K^\bu),
\qquad l=p+q,\leqno(9.3)$$
can be identified with the graded module $G^\bu H^\bu(K^\bu)$ given by the
filtration $F^p H^\bu(K^\bu)$. The terms $E_1$ are defined as the
cohomology groups of the partial complex $d'':K^{p,q}\to
K^{p,q+1}$ with respect to the second differential, that is,
$$E_1^{p,q}=H^q\big((K^{p,\bu},d'')\big),\leqno(9.4)$$
and the differential $d_1:E_1^{p,q}\to E_1^{p+1,q}$ is induced by the first
differential $d'\,:$
$$d':H^q\big((K^{p,\bu},d'')\big)\lra H^q\big((K^{p+1,\bu},d'')\big).
\leqno(9.5)$$
In fact, we have $E_r^{p,q}=0$ unless $p,q\ge 0$, and the limit
$E_\infty=\lim E_r$ is stationary, more precisely
$$E_r^{p,q}=E_{r+1}^{p,q}=\ldots=E_\infty^{p,q}\qquad
\hbox{when $r\ge\max(p+1,q+2)$,}$$
as follows by considering the range in which $d_r$ can be non zero.
One says that the spectral sequence {\it converges} to the graded module of
$H^\bu(K^\bu)$. A careful examination of the terms of small degree leads to
the exact sequence
$$0\lra E^{1,0}_2\lra H^1(K^\bu)\lra E^{0,1}_2\buildo d_2\over\lra E^{2,0}_2
\lra H^2(K^\bu).\leqno(9.6)$$
One says that the spectral sequence {\it degenerates} in $E_{r_0}$ if
$d_r=0$ for all $r\ge r_0$ and for all bidegrees $p,q$. Then we have of course
$E_{r_0}^{\bu,\bu}=E_{r_0+1}^{\bu,\bu}=\ldots=E_\infty^{\bu,\bu}$.

In the case of the Hodge-Fr\"olicher spectral sequence, the terms $E_1$ are
the Dolbeault cohomology groups
$$E^{p,q}_1=H^{p,q}(X,\bbbc).\leqno(9.7)$$
The limit term $E^{p,q}_\infty$ is the graded module associated with a 
filtration $F^pH^l_\DR(X,\bbbc)$, $l=p+q$, of the De Rham cohomology groups.
Since $E_{r+1}=H^\bu(E_r)$, the dimensions $\dim E_r^{p,q}$ are non
increasing with~$r$, hence $\dim E_\infty^{p,q}\le \dim E_r^{p,q}$,
and there is equality if and only if the spectral sequence degenerates
in~$E_r$. In particular, the Betti numbers $b_l=\dim H^l(X,\bbbc)$ and
Hodge numbers $h^{p,q}=\dim E_1^{p,q}$ satisfy the inequality
$$b_l=\sum_{p+q=l}\dim E^{p,q}_\infty\le\sum_{p+q=l}\dim E^{p,q}_1=
\sum_{p+q=l}h^{p,q},\leqno(9.8)$$
and the equality is equivalent to the degeneration of the spectral sequence
at $E^\bu_1$. As a consequence:

\begstat{(9.9) Theorem} If $X$ is a compact complex manifold, the following
properties are equivalent:
\smallskip
\item{\rm a)} The Hodge-Fr\"olicher spectral sequence degenerates at $E^\bu_1$.
\smallskip
\item{\rm b)} There is an equality $b_l=\sum_{p+q=l}h^{p,q}$ for all~$l$.
\smallskip
\item{\rm c)} There is an isomorphism $G^pH^{p+q}_\DR(X,\bbbc)
\simeq H^{p,q}(X,\bbbc)$.
\smallskip
\noindent If one of these conditions is met, isomorphism {\rm c)} is defined
in a canonical way.
\endstat

We may now rephrase the basic results of \S~8.B as follows.

\begstat{(9.10) Theorem} If $X$ is a compact K\"ahler manifold,
the Hodge-Fr\"olicher spectral sequence degenerates at $E_1$ and
there is a canonical decomposition
$$H^l_\DR(X,\bbbc)=\bigoplus_{p+q=l}H^{p,q}(X,\bbbc),\qquad
H^{q,p}(X,\bbbc)=\ovl{H^{p,q}(X,\bbbc)}.$$
In terms of this decomposition, the filtration $F^pH^l_\DR(X,\bbbc)$ is
given by
$$F^pH^l_\DR(X,\bbbc)=\bigoplus_{j\ge p}H^{j,l-j}(X,\bbbc).$$
In particular, the conjugate filtration $\ovl{F^\bu H^l_\DR}$ is opposed to
$F^\bu H^l_\DR$, i.e., 
$$H^l_\DR(X,\bbbc)=F^p H^l_\DR(X,\bbbc)\oplus\ovl{F^{l-p+1}H^l_\DR(X,\bbbc)}.$$
\endstat

\begstat{(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 $E_1$ and if the conjugate filtration
$\ovl{F^\bu H^l_\DR}$ is opposed to $F^\bu H^l_\DR$, i.e.,
$H^l_\DR=F^pH^l_\DR\oplus\ovl{F^{l-p+1}H^l_\DR}$ for all~$p$.
\endstat

If $X$ admits a Hodge decomposition in the sense of Definition 9.11 and
$p+q=l$, we see immediately from the equality
$H^l_\DR=F^{p+1}H^l_\DR\oplus\ovl{F^qH^l_\DR}$ that
$$F^pH^l_\DR=F^{p+1}H^l_\DR\oplus(F^pH^l_\DR\cap\ovl{F^qH^l_\DR}),$$
hence we get a canonical isomorphism
$$H^{p,q}(X,\bbbc)\simeq F^pH^l_\DR/F^{p+1}H^l_\DR\simeq
F^pH^l_\DR\cap\ovl{F^qH^l_\DR}\subset H^l_\DR.\leqno(9.12)$$
We infer from this that there are indeed canonical isomorphisms
$$H^l_\DR(X,\bbbc)=\bigoplus_{p+q=l}H^{p,q}(X),\qquad
H^{q,p}(X,\bbbc)=\ovl{H^{p,q}(X,\bbbc)},$$
as expected. Notice that (9.12) provides another proof of the
fact that the Hodge decomposition of a compact K\"ahler manifold does
not depend on the given K\"ahler metric (all groups and morphisms involved
in (9.12) are intrinsic). In fact, we have proved that compact K\"ahler
manifolds satisfy a still stronger property, which we feel convenient to call
{\it strong Hodge decomposition}, since it trivially implies the existence
of a Hodge decomposition in the sense of Definition~9.11.

\begstat{(9.13) Definition} If $X$ is a compact complex manifold, we say
that $X$ admits a strong Hodge decomposition if 
$$H^{p,q}_\BC(X,\bbbc)\to H^{p,q}(X,\bbbc),\qquad
\bigoplus_{p+q=l}H^{p,q}_\BC(X,\bbbc)\to H^l_\DR(X,\bbbc)$$
are isomorphisms.
\endstat

\begstat{(9.14) Remark} \rm Deligne [Del68, 72] gave a criterion for
proving the degeneration of the Hodge spectral sequence
in a more algebraic setting, covering relative situations. More recently,
Deligne and Illusie [DeI87] gave a proof of the degeneration of the
Hodge spectral sequence without using analytic methods (their idea is to
work in characteristic $p$ and let somehow $p$ tend to $+\infty$). It should
be observed that the degeneration of the Hodge-Fr\"olicher spectral sequence
{\it does not automatically imply} the Hodge symmetry property
$H^{q,p}(X,\bbbc)\simeq\smash{\ovl{H^{p,q}(X,\bbbc)}}$ nor the existence
of a canonical decomposition of the 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~$E_1$; however, if $X$ is
not K\"ahler, then $b_1$ is odd, and one can show by using the Hirzebruch
index theorem that $h^{0,1}=h^{1,0}+1$ and $b_1=2h_{1,0}+1$ (see [BPV84]).
It can be shown that the existence of a Hodge (resp.\ strong Hodge)
decomposition is preserved by blowing down (replacement of $X$ by $X'$,
when $\mu:X\to X'$ is a modification); this is an easy consequence of the
existence of a direct image fonctor $\mu_\star$ acting on all cohomology
groups involved, such that $\mu_\star\mu^\star=\Id\,$; in the analytic
context, $\mu_\star$ is most easily constructed by using the theory of
currents. As all Moishezon manifolds admit projective algebraic
blow-ups, one concludes that these manifolds also have a strong Hodge
decomposition. It would be interesting to know whether there are 
examples of compact complex manifolds possessing a Hodge decomposition
without having a strong Hodge decomposition (there are indeed simple
examples of abstract double complexes with this property).\qed
\endstat

In general, when $X$ is not K\"ahler, interesting informations can be
deduced from the spectral sequence. For instance, (9.6) implies
$$b_1\ge\dim E^{1,0}_2+(\dim E^{0,1}_2-\dim E^{2,0}_2)_+.\leqno(9.15)$$
However, $E^{1,0}_2$ is the central cohomology group in the sequence
$$d_1=d':E^{0,0}_1\lra E^{1,0}_1\lra E^{2,0}_1,$$
and as $E^{0,0}_1$ is the space of holomorphic functions on $X$, the
first map $d_1$ is zero (by the maximum principle, holomorphic functions
are constant on each connected component of~$X$). Hence
$\dim E^{1,0}_2\ge h^{1,0}-h^{2,0}$. Similarly, $E^{0,1}_2$ is
the kernel of a map $E^{0,1}_1\to E^{1,1}_1$, thus
$\dim E^{0,1}_2\ge h^{0,1}-h^{1,1}$. By (9.15) we obtain
$$b_1\ge (h^{1,0}-h^{2,0})_++(h^{0,1}-h^{1,1}-h^{2,0})_+.\leqno(9.16)$$
Another interesting relation concerns the topological Euler-Poincar\'e
characteristic
$$\chi_{{\rm top}}(X)=b_0-b_1+\ldots-b_{2n-1}+b_{2n}.$$
We need the following simple lemma.

\begstat{(9.17) Lemma} Let $(C^\bu,d)$ a bounded complex of finite dimensional
vector spaces over some field. Then, the Euler characteristic 
$$\chi(C^\bu)=\sum(-1)^q\dim C^q$$
is equal to the Euler characteristic $\chi\big(H^\bu(C^\bu)\big)$ of the
cohomology module. 
\endstat

\begproof{} Set
$$c_q=\dim C^q,\quad z_q=\dim Z^q(C^\bu),\quad b_q=\dim B^q(C^\bu),\quad
h_q=\dim H^q(C^\bu).$$
Then
$$c_q=z_q+b_{q+1},\qquad h_q=z_q-b_q.$$
Therefore we find
$$\sum(-1)^q\,c_q=\sum(-1)^q\,z_q-\sum(-1)^q\,b_q=\sum(-1)^q\,h_q.
\eqno{\square}$$
\endproof

In particular, if the term $E^\bu_r$ of the spectral sequence of a
filtered complex $K^\bu$ is a bounded and finite dimensional complex, we 
have
$$\chi(E^\bu_r)=\chi(E^\bu_{r+1})=\ldots=\chi(E^\bu_\infty)=
\chi\big(H^\bu(K^\bu)\big)$$
because $E^\bu_{r+1}=H^\bu(E^\bu_r)$ and $\dim E^l_\infty=\dim H^l(K^\bu)$.
In the Hodge-Fr\"olicher spectral sequence, we have
$\dim E^l_1=\sum_{p+q=l}h^{p,q}$, hence:

\begstat{(9.18) Theorem} For any compact complex manifold $X$, one has
$$\chi_{{\rm top}}(X)=\sum_{0\le k\le 2n}(-1)^lb_l=\sum_{0\le p,q\le n}
(-1)^{p+q}h^{p,q}.$$
\endstat
\vfill\eject

\titleb{\hfil\bgroup\titleafonts Second Part: $L^2$ Vanishing Theorems
\egroup\hfil}{}
\titlearunning{Second Part: $L^2$ Vanishing Theorems}
\titleb{\S 10.}{Pseudoconvexity and Positivity Concepts}
Several pseudoconvexity and positivity notions are involved in the
statements and proofs of vanishing theorems. We give here a review of the
most basic concepts needed.

\titlec{\S 10.A.}{Plurisubharmonic Functions}
Plurisubharmonic functions have been introduced independently
by Lelong and Oka in the study of holomorphic convexity. We refer to
[Lel67, 69] for more details.

\begstat{(10.1) Definition} A function $u:\Omega\lra[-\infty,+\infty[$ defined
on an open subset $\Omega\subset\bbbc^n$ is said to be plurisubharmonic
$($psh for short$)$ if
\smallskip
\item{\rm a)} $u$ is upper semicontinuous~$;$
\smallskip
\item{\rm b)} for every complex line $L\subset\bbbc^n$, 
$u_{\restriction\Omega\cap L}$ is subharmonic on $\Omega\cap L$, that is,
for all $a\in\Omega$ and $\xi\in\bbbc^n$ with $|\xi|<d(a,\complement\Omega)$,
the function $u$ satisfies the mean value inequality
$$u(a)\le{1\over 2\pi}\int_0^{2\pi}u(a+e^{\ii\theta}\,\xi)\,d\theta.$$
\vskip0pt\noindent
The set of psh functions on $\Omega$ is denoted by
$\Psh(\Omega)$.
\endstat

We list below the most basic properties of psh functions.
They all follow easily from the definition.

\begstat{(10.2) Basic properties} \rm
\smallskip
\item{\rm a)} Every function $u\in\Psh(\Omega)$ is subharmonic, namely it
satisfies the mean value inequality on euclidean balls or spheres:
$$u(a)\le{1\over\pi^nr^{2n}/n!}\int_{B(a,r)}u(z)\,d\lambda(z)$$
for every $a\in\Omega$ and $r<d(a,\complement\Omega$. Either $u\equiv-\infty$
or $u\in L^1_\loc$ on every connected component of~$\Omega$.
\smallskip
\item{\rm b)} For any decreasing sequence of psh
functions $u_k\in\Psh(\Omega)$, the limit $u=\lim u_k$ is psh
on $\Omega$.
\smallskip
\item{\rm c)} Let $u\in\Psh(\Omega)$ be such that $u\not\equiv-\infty$ 
on every connected component of~$\Omega$. If $(\rho_\varepsilon)$ is a 
family of smoothing kernels, then $u\star\rho_\varepsilon$ is $\ci$ and 
psh on
$$\Omega_\varepsilon=\big\{x\in\Omega\,;\,d(x,\complement\Omega)>\varepsilon
\big\},$$
the family $(u\star\rho_\varepsilon)$ is increasing in $\varepsilon$ and 
$\lim_{\varepsilon\to 0}u\star\rho_\varepsilon=u$.
\smallskip
\item{\rm d)} Let $u_1\ld u_p\in\Psh(\Omega)$ and
$\chi:\bbbr^p\lra\bbbr$ be a convex function such that $\chi(t_1\ld t_p)$
is increasing in each $t_j$. Then $\chi(u_1\ld u_p)$ is psh
on $\Omega$. In particular~ $u_1+\cdots+u_p$, $\max\{u_1\ld u_p\}$, 
$\log(e^{u_1}+\cdots+e^{u_p})$ are psh on~$\Omega$.\qed
\smallskip
\endstat

\begstat{(10.3) Lemma} A function $u\in C^2(\Omega,\bbbr)$ is psh on
$\Omega$ if and only if the hermitian form $Hu(a)(\xi)=\sum_{1\le j,k\le n}
\partial^2 u/\partial z_j\partial\ovl z_k(a)\,\xi_j\ovl \xi_k$
is semipositive at every point $a\in\Omega$.
\endstat

\begproof{} This is an easy consequence of the following standard formula
$${1\over 2\pi}\int_0^{2\pi}u(a+e^{\ii\theta}\,\xi)\,d\theta - u(a)=
{2\over\pi}\int_0^1{dt\over t}\int_{|\zeta|<t}Hu(a+\zeta\xi)(\xi)\,
d\lambda(\zeta),$$
where $d\lambda$ is the Lebesgue measure on $\bbbc$.
Lemma~10.3 is a strong evidence that plurisubharmonicity is the natural
complex analogue of linear convexity.\qed
\endproof

For non smooth functions, a similar characterization of plurisubharmonicity
can be obtained by means of a regularization process.

\begstat{(10.4) Theorem} If $u\in\Psh(\Omega)$, $u\not\equiv-\infty$ on every
connected component of $\Omega$, then for all $\xi\in\bbbc^n$
$$Hu(\xi)=\sum_{1\le j,k\le n}{\partial^2 u\over\partial z_j\partial\ovl z_k}
\,\xi_j\ovl\xi_k\in\cD'(\Omega)$$
is a positive measure. Conversely, if $v\in\cD'(\Omega)$ is such that 
$Hv(\xi)$ is a positive measure for every $\xi\in\bbbc^n$, there exists a 
unique function $u\in\Psh(\Omega)$ which is locally integrable on $\Omega$
and such that $v$ is the distribution associated to $u$.\qed
\endstat

In order to get a better geometric insight of this notion, we assume more
generally that $u$ is a function on a complex $n$-dimensional manifold 
$X$. If $\Phi:X\to Y$ is a holomorphic mapping and if $v\in C^2(Y,\bbbr)$, we
have $\hbox{$d'd''(v\circ \Phi)$}=\Phi^\star d'd''v$, hence
$$H(v\circ \Phi)(a,\xi)=Hv\big(\Phi(a),\Phi'(a).\xi\big).$$
In particular $Hu$, viewed as a hermitian form on $T_X$, does not depend
on the choice of coordinates $(z_1\ld z_n)$. Therefore, the notion of
psh function makes sense on any complex manifold. More generally,
we have

\begstat{(10.5) Proposition} If $\Phi:X\lra Y$ is a holomorphic map and
$v\in\Psh(Y)$, then $v\circ \Phi\in\Psh(X)$.\qed
\endstat

\begstat{(10.6) Example} \rm It is a standard fact that $\log|z|$ is psh
(i.e.\ subharmonic) on $\bbbc$. Thus $\log|f|\in\Psh(X)$ for every
holomorphic function $f\in H^0(X,\cO_X)$. More generally 
$$\log\big(|f_1|^{\alpha_1}+\cdots+|f_q|^{\alpha_q}\big)\in\Psh(X)$$
for every $f_j\in H^0(X,\cO_X)$ and $\alpha_j\ge 0$ (apply Property~10.2$\,$d
with $u_j=\alpha_j\,\log|f_j|$). We will be especially interested in the
singularities obtained at points of the zero variety $f_1=\ldots=f_q=0$,
when the $\alpha_j$ are rational numbers.\qed
\endstat

\begstat{(10.7) Definition} A psh function $u\in\Psh(X)$ will be said to
have algebraic singularities if $u$ can be written locally as
$$u={\alpha\over 2}\log\big(|f_1|^2+\cdots+|f_N|^2\big)+v,$$
with holomorphic functions $f_j$, $\alpha\in\bbbq$, and $v$ a
bounded function.
\endstat

We then introduce the ideal $\cJ=\cJ(u/\alpha)$ of germs of
holomorphic functions $h$ such that $|h|\le Ce^{u/\alpha}$ for some constant
$C$, i.e.
$$|h|\le C\big(|f_1|+\cdots+|f_N|\big).$$
This is a globally defined ideal sheaf on $X$, locally equal to the
integral closure $\ovl\cI$ of the ideal sheaf $\cI=(f_1\ld f_N)$, thus $\cJ$
is coherent on~$X$. If $(g_1\ld g_{N'})$ are local generators of $\cJ$,
then we still have
$$u={\alpha\over 2}\log\big(|g_1|^2+\cdots+|g_{N'}|^2\big)+O(1),$$
From an algebraic point of view, the singularities of $u$ are in 1:1
correspondence with the ``algebraic data'' $(\cJ,\alpha)$. Later
on, we will see another important method for associating an ideal sheaf
to a psh function.

\titlec{10.B.}{Positive Currents}
The theory of currents has been introduced by G.~de Rham [DR55].
We recall here only the most basic definitions. The reader can
consult [Fed69] for a more thorough treatment of current theory.
In the complex situation, there is an important concept of positive
current, defined and developed by [Lel57, 69].

A {\it current} of degree $q$ on an oriented differentiable manifold $M$ is
simply a differential $q$-form $\Theta$ with distribution coefficients.
The space of currents of degree $q$ over $M$ will be denoted by
$\cD^{\prime q}(M)$. Alternatively, a current of degree $q$ can be seen
as an element $\Theta$ in the dual space $\cD'_p(M):=\big(\cD^p(M)\big)'$ of
the space $\cD^p(M)$ of smooth differential forms of degree $p=\dim M-q$ with
compact support; the duality pairing is given by
$$\langle \Theta,\alpha\rangle=\int_M \Theta\wedge\alpha,~~~~
\alpha\in\cD^p(M).\leqno(10.8)$$
A basic example is the {\it current of integration} $[S]$ over a compact
oriented submanifold $S$ of $M\,$:
$$\langle[S],\alpha\rangle=\int_S\alpha,~~~~{\rm deg}\alpha=p=\dim_\bbbr S.
\leqno(10.9)$$
Then $[S]$ is a current with measure coefficients, and
Stokes' formula shows that $d[S]=(-1)^{q-1}[\partial S]$, in particular
$d[S]=0$ if $S$ has no boundary. Because of this example, the integer $p$
is said to be the dimension of~$\Theta$ when $\Theta\in\cD_p'(M)$. The
current $\Theta$ is said to be {\it closed} if $d\Theta=0$.

On a complex manifold $X$, we have similar notions of bidegree and
bidimension; as in the real case, we denote by
$$\cD^{\prime p,q}(X)=\cD'_{n-p,n-q}(X),\qquad n=\dim X,$$
the space of currents of bidegree $(p,q)$ and bidimension $(n-p,n-q)$ on~$X$.
According to [Lel57], a current $\Theta$
of bidimension $(p,p)$ is said to be {\it $($weakly$)$ positive} if
for every choice of smooth $(1,0)$-forms $\alpha_1\ld\alpha_p$ on~$X$
the distribution
$$\Theta\wedge\ii\alpha_1\wedge\ovl\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\ovl\alpha_p~~~~\hbox{\rm is a positive measure.}
\leqno(10.10)$$

\begstat{(10.11) Exercise} \rm If $\Theta$ is positive, show that the
coefficients $\Theta_{I,J}$ of $\Theta$ are complex measures, and that,
up to constants, they are dominated by the trace measure
$$\sigma_\Theta=\Theta\wedge{1\over p!}\beta^p=2^{-p}\sum\Theta_{I,I},\qquad
\beta={\ii\over 2}d'd''|z|^2={\ii\over 2}\sum_{1\le j\le n}dz_j\wedge
d\ovl z_j,$$
which is a positive measure.\newline
{\it Hint}. Observe that $\sum\Theta_{I,I}$ is invariant by unitary changes
of coordinates and that the $(p,p)$-forms
$\ii\alpha_1\wedge\ovl\alpha_1\wedge\ldots\wedge\ii\alpha_p\wedge\ovl\alpha_p$
generate $\Lambda^{p,p}T^\star_{\bbbc^n}$ as a $\bbbc$-vector space.\qed
\endstat

A current $\Theta=\ii\sum_{1\le j,k\le n}\Theta_{jk}dz_j\wedge dz_k$
of bidegree $(1,1)$ is easily seen to be positive if and only if the
complex measure $\sum\lambda_j\ovl\lambda_k \Theta_{jk}$ is a positive
measure for every $n$-tuple $(\lambda_1\ld\lambda_n)\in\bbbc^n$.

\begstat{(10.12) Example} \rm If $u$ is a (not identically $-\infty$)
psh function on $X$, we can associate with $u$ a (closed) positive
current $\Theta=\ii\partial\ovl\partial u$ of bidegree~$(1,1)$. Conversely,
every closed positive current of bidegree $(1,1)$ can be written under
this form on any open subset $\Omega\subset X$ such that
$H^2_{DR}(\Omega,\bbbr)=H^1(\Omega,\cO)=0$, e.g.\ on small coordinate
balls (exercise to the reader).\qed
\endstat

It is not difficult to show that a product $\Theta_1\wedge\ldots\wedge
\Theta_q$ of positive currents of bidegree $(1,1)$ is positive whenever
the product is well defined (this is certainly the case if all
$\Theta_j$ but one at most are smooth; much finer conditions will be
discussed in Section~2).

We now discuss another very important example of closed positive current.
In fact, with every closed analytic set $A\subset X$ of pure dimension $p$
is associated a current of integration
$$\langle [A],\alpha\rangle=\int_{A_\reg}\alpha,~~~~\alpha\in\cD^{p,p}(X),
\leqno(10.13)$$
obtained by integrating over the regular points of~$A$. In order to show
that (10.13) is a correct definition of a current on~$X$, one must show that
$A_\reg$ has locally finite area in a neighborhood of $A_\sing$.
This result, due to [Lel57] is shown as follows. Suppose that $0$
is a singular point of $A$. By the local parametrization theorem for
analytic sets, there is a linear change of coordinates on $\bbbc^n$
such that all projections
$$\pi_I:(z_1\ld z_n)\mapsto (z_{i_1}\ld z_{i_p})$$
define a finite ramified covering of the intersection $A\cap\Delta$ with
a small polydisk $\Delta$ in $\bbbc^n$ onto a small
polydisk $\Delta_I$ in~$\bbbc^p$. Let $n_I$ be the sheet number. Then the
\hbox{$p$-dimensional} area of $A\cap\Delta$ is bounded above by the sum
of the areas of its projections counted with multiplicities, i.e.
$$\hbox{\rm Area}(A\cap\Delta)\le\sum n_I\hbox{\rm Vol}(\Delta_I).$$
The fact that $[A]$ is positive is also easy. In fact
$$\ii\alpha_1\wedge\ovl\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\ovl\alpha_p=|\det(\alpha_{jk})|^2\,
\ii w_1\wedge\ovl w_1\wedge\ldots\wedge
\ii w_p\wedge\ovl w_p$$
if $\alpha_j=\sum\alpha_{jk}dw_k$ in terms of local coordinates
$(w_1\ld w_p)$ on $A_\reg$. This shows that all such forms are $\ge 0$
in the canonical orientation defined by
$\ii w_1\wedge\ovl w_1\wedge\ldots\wedge\ii w_p\wedge\ovl w_p$.
More importantly, Lelong [Lel57] has shown that $[A]$ is $d$-closed in~$X$,
even at points of $A_\sing$. Now, in relation with Example~10.12,
we have the important

\vbox{\begstat{(10.14) Lelong-Poincar\'e equation}
Let $f\in H^0(X,\cO_X)$ be a non zero holomorphic function,
$Z_f=\sum m_jZ_j$, $m_j\in\bbbn$, the zero divisor of~$f$ and
$[Z_f]=\sum m_j[Z_j]$ the associated current of integration. Then
$${\ii\over\pi}\partial\ovl\partial\log|f|=[Z_f].$$
\endstat\removelastskip}

\begproof{(sketch).} It is clear that $\ii d'd''\log|f|=0$ in a neighborhood
of every point $x\notin\Supp(Z_f)=\bigcup Z_j$, so it is enough to check
the equation in a neighborhood of every point of $\Supp(Z_f)$. Let $A$ be
the set of singular points of $\Supp(Z_f)$, i.e.\ the union of the
pairwise intersections $Z_j\cap Z_k$ and of the singular loci
$Z_{j,\sing}$; we thus have $\dim A\le n-2$. In a neighborhood of any
point $x\in\Supp(Z_f)\ssm A$ there are local coordinates $(z_1\ld z_n)$
such that $f(z)=z_1^{m_j}$ where $m_j$ is the multiplicity of $f$ along
the component $Z_j$ which contains~$x$ and $z_1=0$ is an equation for
$Z_j$ near~$x$. As ${\ii\over\pi}d'd''\log|z|={}$ Dirac measure $\delta_0$
in $\bbbc$, we find ${\ii\over\pi}d'd''\log|z_1|=[$hyperplane $z_1=0]$,
hence
$${\ii\over\pi}d'd''\log|f|=m_j{\ii\over\pi}d'd''\log|z_1|=m_j[Z_j]$$
in a neighborhood of $x$, as desired. This shows that the equation holds
on $X\ssm A$. Hence the difference ${\ii\over\pi}d'd''\log|f|-[Z_f]$ is
a closed current of degree $2$ with measure coefficients, whose support
is contained in~$A$. This current must be $0$, for $A$ has too small
dimension to carry its support ($A$ is stratified by submanifolds of
real codimension $\ge 4$, whilst the current itself has real
codimension~$2$).\qed
\endproof

To conclude this section, we discuss De Rham and Dolbeault cohomology theory
in the context of currents. A basic observation is that the Poincar\'e and
Dolbeault-Grothendieck lemma still hold for currents. Namely, if
$(\cD^{\prime q},d)$ and $(\cD'(F)^{p,q},d'')$ denote the complex of sheaves
of degree $q$ currents (resp.\ of $(p,q)$-currents with values in a
holomorphic vector bundle~$F$), we still have De Rham and Dolbeault sheaf
resolutions
$$0\to\bbbr\to\cD^{\prime \bu},\qquad 0\to\Omega^p_X\otimes\cO(F)\to
\cD'(F)^{p,\bu}.$$
Hence we get canonical isomorphisms
$$\leqalignno{
H^q_\DR(M,\bbbr)&=H^q\big((\Gamma(M,\cD^{\prime\bu}),d)\big),&(10.15)\cr
H^{p,q}(X,F)&=H^q\big((\Gamma(X,\cD'(F)^{p,\bu}),d'')\big).&\cr}$$
In other words, we can attach a cohomology class
$\{\Theta\}\in H^q_\DR(M,\bbbr)$ to any closed current $\Theta$ of degree~$q$,
resp.\ a cohomology class $\{\Theta\}\in H^{p,q}(X,F)$ to any $d''$-closed
current of bidegree~$(p,q)$. Replacing if necessary every current by a
smooth representative in the same cohomology class, we see that there is a
well defined cup product given by the wedge product of differential forms
$$\eqalign{
H^{q_1}(M,\bbbr)\times\ldots\times H^{q_m}(M,\bbbr)&\lra
H^{q_1+\ldots+q_m}(M,\bbbr),\cr
(\{\Theta_1\}\ld\{\Theta_1\})&\longmapsto\{\Theta_1\}\wedge\ldots\wedge
\{\Theta_m\}.\cr}$$
In particular, if $M$ is a compact oriented variety and
$q_1+\ldots+q_m=\dim M$, there is a well defined intersection number
$$\{\Theta_1\}\cdot\{\Theta_2\}\cdot\,\cdots\,\cdot
\{\Theta_m\}=\int_M\{\Theta_1\}\wedge\ldots\wedge\{\Theta_m\}.$$
Notice, however, that the pointwise product
\hbox{$\Theta_1\wedge\ldots\wedge\Theta_m$} need not exist in general.

\titlec{\S 10.C.}{Positive Vector Bundles}
Let $(E,h)$ be a holomorphic hermitian vector bundle on a complex
manifold~$X$. Its Chern curvature tensor
$$\ii\Theta(E)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}dz_j\wedge dz_k\otimes e_\lambda^\star\otimes e_\mu.$$
can be identified with a hermitian form on $T_X\otimes E$,
$$\wt\Theta(E)(\xi\otimes v)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\xi_j\ovl\xi_kv_\lambda\ovl v_\mu,\qquad
\ovl c_{jk\lambda\mu}=c_{kj\mu\lambda}.\leqno(1.11)$$
This leads in a natural way to positivity concepts, following definitions
introduced by Kodaira [Kod53], Nakano [Nak55] and Griffiths [Gri66].

\begstat{(10.16) Definition} The hermitian vector bundle $E$ is said to be
\smallskip
\item{\rm a)} positive in the sense of Nakano if$\,:$\hfill\break
$\wt\Theta(E)(\tau)>0$ for all non zero tensors $\tau=\sum
\tau_{j\lambda}\partial/\partial z_j\otimes e_\lambda\in T_X\otimes E$.
\smallskip
\item{\rm b)} positive in the sense of Griffiths if$\,:$\hfill\break
$\wt\Theta(E)(\xi\otimes v)>0$ for all non zero decomposable tensors
$\xi\otimes v\in T_X\otimes E\,;$
\smallskip\noindent
Corresponding semipositivity concepts are defined by relaxing the strict
inequalities.
\endstat

\begstat{(10.17) Special case of rank 1 bundles} \rm Assume that
$E$ is a line bundle. The hermitian matrix $H=(h_{11})$ associated
to a trivialization $\tau:E_{\restriction\Omega}\simeq\Omega\times\bbbc$
is simply a positive function which we find convenient to denote by
$e^{-2\varphi}$, \hbox{$\varphi\in C^\infty(\Omega,\bbbr)$}. In this case
the curvature form $\Theta(E)$ can be identified to the $(1,1)$-form
$2d'd''\varphi$, and
$${\ii\over 2\pi}\Theta(E)={\ii\over\pi}d'd''\varphi=dd^c\varphi$$
is a real $(1,1)$-form. Hence $E$ is semipositive (in either Nakano or
Griffiths sense) if and only if $\varphi$ is psh, resp.\ positive if
and only if $\varphi$ is {\it strictly psh}. In this setting, the
Lelong-Poincar\'e equation can be generalized as follows: let
$\sigma\in H^0(X,E)$ be a non zero holomorphic section. Then
$$dd^c\log\|\sigma\|=[Z_\sigma]-{\ii\over 2\pi}\Theta(E).\leqno(10.18)$$
Formula (10.18) is immediate if we write $\|\sigma\|=|\tau(\sigma)|
e^{-\varphi}$ and if we apply the Lelong-Poincar\'e equation to the
holomorphic function $f=\tau(\sigma)$. As we shall see later, it is
very important for applications to consider also singular hermitian metrics.
\endstat

\begstat{(10.19) Definition} A singular $($hermitian$)$ metric on a line
bundle $E$ is a metric which is given in any trivialization
$\tau:E_{\restriction\Omega}
\buildo\simeq\over\longrightarrow\Omega\times\bbbc$ by
$$\|\xi\|=|\tau(\xi)|\,e^{-\varphi(x)},~~~~x\in\Omega,~\xi\in E_x$$
where $\varphi\in L^1_\loc(\Omega)$ is an arbitrary function, called the
weight of the metric with respect to the trivialization $\tau$.
\endstat

If $\tau':E_{\restriction\Omega'}\lra\Omega'\times\bbbc$ 
is another trivialization, $\varphi'$ the associated weight and
$g\in\cO^\star(\Omega\cap\Omega')$ the transition function, then
$\tau'(\xi)=g(x)\,\tau(\xi)$ for $\xi\in E_x$, and so
$\varphi'=\varphi+\log|g|$ on $\Omega\cap\Omega'$. The curvature form 
of $E$ is then given formally by the closed $(1,1)$-current
${\ii\over\pi}\Theta(E)=dd^c\varphi$ on $\Omega\,$; our assumption
$\varphi\in L^1_\loc(\Omega)$ guarantees that $\Theta(E)$ exists 
in the sense of distribution theory. As in the smooth case, ${\ii\over\pi}
\Theta(E)$ 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 $c_1(E)\in H^2(X,\bbbz)$ in $H^2_{DR}(X,\bbbr)$.
Before going further, we discuss two basic examples. 

\begstat{(10.20) Example} \rm Let $D=\sum\alpha_jD_j$ be a divisor with 
coefficients $\alpha_j\in\bbbz$ and let $E=\cO(D)$ be the associated
invertible sheaf of meromorphic functions $u$ such that
${\rm div}(u)+D\ge 0\,$; the corresponding line bundle can be
equipped with the singular metric defined by $\|u\|=|u|$. If $g_j$ is a
generator of the ideal of $D_j$ on an open set $\Omega\subset X$ then 
$\tau(u)=u\prod\smash{g_j^{\alpha_j}}$ defines a trivialization of
$\cO(D)$ over $\Omega$, thus our singular metric is associated to the
weight $\varphi=\sum\alpha_j\log|g_j|$. By the Lelong-Poincar\'e equation,
we find
$${\ii\over\pi}\Theta\big(\cO(D)\big)=dd^c\varphi=[D],$$
where $[D]=\sum\alpha_j[D_j]$ denotes the current of integration
over~$D$.\qed
\endstat

\begstat{(10.21) Example} \rm Assume that $\sigma_1\ld \sigma_N$ are non zero 
holomorphic sections of $E$. Then we can define a natural (possibly
singular) hermitian metric 
on $E^\star$ by
$$\|\xi^\star\|^2=\sum_{1\le j\le n}\big|\xi^\star.\sigma_j(x)\big|^2
~~~~\hbox{\rm for}~~\xi^\star\in E^\star_x.$$
The dual metric on $E$ is given by
$$\|\xi\|^2={|\tau(\xi)|^2\over|\tau(\sigma_1(x))|^2+\ldots+
|\tau(\sigma_N(x))|^2}$$
with respect to any trivialization $\tau$. The associated weight function is
thus given by $\varphi(x)=\log\big(\sum_{1\le j\le N}|\tau(\sigma_j(x))|^2
\big){}^{1/2}$. In this case $\varphi$ is a psh function, thus
$\ii\Theta(E)$ is a closed positive current. Let us denote by $\Sigma$
the linear system defined by $\sigma_1\ld \sigma_N$ and by
$B_\Sigma=\bigcap\sigma_j^{-1}(0)$ its base locus. We have a meromorphic
map
$$\Phi_\Sigma:X\ssm B_\Sigma\to\bbbp^{N-1},\qquad
x\mapsto(\sigma_1(x):\sigma_2(x):\ldots:\sigma_N(x)).$$
Then ${\ii\over\pi}\Theta(E)$ is equal to the pull-back over $X\ssm B_\Sigma$
of the Fubini-Study metric $\omega_\FS={\ii\over 2\pi}\log(|z_1|^2+\ldots+
|z_N|^2)$ of $\bbbp^{N-1}$ by $\Phi_\Sigma$ .\qed
\endstat

\begstat{(10.22) Ample and very ample line bundles} A holomorphic line
bundle $E$ over a compact complex manifold $X$ is said to be
\smallskip
\item{\rm a)} very ample if the map $\Phi_{|E|}:X\to\bbbp^{N-1}$ associated
with the complete linear system \hbox{$|E|=P(H^0(X,E))$} is a regular embedding
$($by this we mean in particular that the base locus is empty, i.e.\
$B_{|E|}=\emptyset)$.
\smallskip
\item{\rm b)} ample if some multiple $mE$, $m>0$, is very ample.
\endstat

Here we use an additive notation for $\Pic(X)=H^1(X,\cO^\star)$, hence
the symbol $mE$ denotes the line bundle $E^{\otimes m}$. By
Example~10.22, every ample line bundle $E$ has a smooth hermitian metric
with positive definite curvature form; indeed, if the linear system
$|mE|$ gives an embedding in projective space, then we get a smooth
hermitian metric on $E^{\otimes m}$, and the $m$-th root yields a
metric on $E$ such that ${\ii\over 2\pi}\Theta(E)={1\over m}
\Phi_{|mE|}^\star\omega_\FS$. Conversely, the Kodaira embedding theorem
[Kod54] tells us that every positive line bundle $E$ is ample (see
Exercise~13.11 for a straightforward analytic proof of the Kodaira
embedding theorem).

\titleb{\S 11.}{Bochner Technique and Vanishing Theorems}
Let $X$ be a compact complex manifold equipped with a K\"ahler metric
$\omega=\sum\omega_{jk}dz_j\wedge d\ovl z_k$. Let $(E,h)$ be a holomorphic
hermitian vector bundle on $X$. We denote by $D=D'+D''$ its Chern
connection and by $\Theta(E)$ its curvature tensor.

\begstat{(11.1) Basic commutation relations} Let $L$ be the operators
defined by $Lu=\omega\wedge u$ and $\Lambda=L^\star$ as usual. Then
$$\eqalign{
[D^{\prime\prime\star},L]&=\ii D',\cr
[\Lambda,D'']&=-\ii D^{\prime\star},\cr}
\qquad
\eqalign{
[D^{\prime\star},L]&=-\ii D'',\cr
[\Lambda,D']&=\ii D^{\prime\prime\star}.\cr}$$
\endstat

\begproof{(sketch).} This is a rather simple consequence of the
commutation relations (6.14) already proved for the trivial connection
$d=d'+d''$ on $E=X\times\bbbc$. In fact, for every point $x_0\in X$,
there is a local holomorphic frame $(e_\lambda)_{1\le\lambda\le r}$
of $E$ such that
$$\langle e_\lambda,e_\mu\rangle=\delta_{\lambda\mu}+O(|z|^2).$$
(The proof is entirely similar to the proof of Theorem~5.8). Then
for $s=\sum s_\lambda\otimes e_\lambda$, $s_\lambda\in C^\infty(X,
\Lambda^{p,q}T^\star_X$, we get
$$D''s=\sum d''s_\lambda\otimes e_\lambda+O(|z|),\qquad
D^{\prime\prime\star}s=\sum d^{\prime\prime\star}
s_\lambda\otimes e_\lambda+O(|z|).$$
Our claim follows.\qed
\endproof

\begstat{(11.2) Bochner-Kodaira-Nakano identity} If $(X,\omega)$ is
K\"ahler, the complex Laplace operators $\Delta'$ and $\Delta''$ acting
on $E$-valued forms satisfy the identity
$$\Delta''=\Delta'+[\ii\Theta(E),\Lambda].$$
\endstat

\begproof{} The last equality in (11.1) yields $D^{\prime\prime\star}
=-\ii[\Lambda,D']$, hence
$$\Delta''=[D'',\delta'']=-\ii[D'',\big[\Lambda,D']\big].$$
By the Jacobi identity we get
$$\big[D'',[\Lambda,D']\big]=\big[\Lambda,[D',D'']]+\big[D',[D'',\Lambda]\big]
=[\Lambda,\Theta(E)]+\ii[D',D^{\prime\star}],$$
taking into account that $[D',D'']=D^2=\Theta(E)$. The formula follows.\qed
\endproof

Assume that $X$ is compact and that $u\in\ci(X,\Lambda^{p,q}T^\star
X\otimes E)$ is an arbitrary $(p,q)$-form. An integration by parts yields
$$\langle\Delta'u,u\rangle=\|D'u\|^2+\|D^{\prime\star}u\|^2\ge0$$
and similarly for $\Delta''$, hence we get the basic a priori inequality
$$\|D''u\|^2+\|D^{\prime\prime\star}u\|^2\ge\int_X
\langle[\ii\Theta(E),\Lambda]u,u\rangle dV_\omega.\leqno(11.3)$$
This inequality is known as the {\it Bochner-Kodaira-Nakano} inequality
(see [Boc48], [Kod53], [Nak55]). When $u$ is $\Delta''$-harmonic, we get
$$\int_X\langle[\ii\Theta(E),\Lambda]u,u\rangle\,dV\le 0.$$
If the hermitian operator $[\ii\Theta(E),\Lambda]$ acting on
$\Lambda^{p,q}T^\star_X\otimes E$ is positive on each fibre, we infer
that $u$ must be zero, hence
$$H^{p,q}(X,E)=\cH^{p,q}(X,E)=0$$
by Hodge theory. The main point is thus to compute the curvature form
$\Theta(E)$ and find sufficient conditions under which the operator
$[\ii\Theta(E),\Lambda]$ is positive definite. Elementary (but somewhat
tedious) calculations yield the following formulae: if the curvature of
$E$ is written as in (1.11) and 
$u=\sum u_{J,K,\lambda}dz_I\wedge d\ovl z_J\otimes e_\lambda$,
$|J|=p$, $|K|=q$, $1\le\lambda\le r$ is a $(p,q)$-form with values in~$E$,
then
$$\leqalignno{\langle[\ii\Theta(E),\Lambda]u,u\rangle=
&\phantom{{}+{}}\sum_{j,k,\lambda,\mu,J,S}{\,}
c_{jk\lambda\mu}\,u_{J,jS,\lambda}~\ovl{u_{J,kS,\mu}}&(11.4)\cr
&{}+\sum_{j,k,\lambda,\mu,R,K}
c_{jk\lambda\mu}\,u_{kR,K,\lambda}~\ovl{u_{jR,K,\mu}}\cr
&{}-{\,}\sum_{j,\lambda,\mu,J,K}\phantom{k}
c_{jj\lambda\mu}\,u_{J,K,\lambda}~\ovl{u_{J,K,\mu}},\cr}$$
where the sum is extended to all indices $1\le j,k\le n$, $1\le\lambda,\mu\le
r$ and multiindices $|R|=p-1$, $|S|=q-1$ (here the notation
$u_{JK\lambda}$ is extended to non necessarily increasing multiindices by
making it alternate with respect to permutations). It is usually hard to
decide the sign of the curvature term (11.4), except in some special cases.

The easiest case is when $p=n$. Then all terms in the second summation of
(11.4) must have $j=k$ and $R=\{1\ld n\}\ssm\{j\}$, therefore the second
and third summations are equal. It follows that $[\ii\Theta(E),\Lambda]$ is
positive on $(n,q)$-forms under the assumption that $E$ is positive in
the sense of Nakano. In this case $X$ is automatically K\"ahler since
$$\omega=\Tr_E(\ii\Theta(E))=\ii\sum_{j,k,\lambda}c_{jk\lambda\lambda}dz_j
\wedge d\ovl z_k=\ii\Theta(\det E)$$
is a K\"ahler metric.

\begstat{(11.5) Nakano vanishing theorem {\rm (1955)}} Let $X$ be a
compact complex manifold and let $E$ be a Nakano positive vector
bundle on~$X$. Then
$$H^{n,q}(X,E)=H^q(X,K_X\otimes E)=0\qquad\hbox{for every $q\ge 1$.}
\eqno\square$$
\endstat

Another tractable case is the case where $E$ is a line bundle ($r=1$).
Indeed, at each point $x\in X$, we may then choose a coordinate
system which diagonalizes simultaneously the hermitian forms
$\omega(x)$ and $\Theta(E)(x)$, in such a way that
$$\omega(x)=\ii\sum_{1\le j\le n}dz_j\wedge d\ovl z_j,\qquad
\Theta(E)(x)=\ii\sum_{1\le j\le n}\gamma_jdz_j\wedge d\ovl z_j$$
with $\gamma_1\le\ldots\le\gamma_n$. The curvature eigenvalues
$\gamma_j=\gamma_j(x)$ are then uniquely defined and depend continuously
on~$x$. With our previous notation, we have $\gamma_j=c_{jj11}$ and all
other coefficients $c_{jk\lambda\mu}$ are zero. For any $(p,q)$-form
$u=\sum u_{JK}dz_J\wedge d\ovl z_K\otimes e_1$, this gives
$$\leqalignno{\langle[\ii\Theta(E),\Lambda]u,u\rangle
&=\sum_{|J|=p,\,|K|=q}\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}\gamma_j-
\sum_{1\le j\le n}\gamma_j\Big)|u_{JK}|^2\cr
&\ge(\gamma_1+\ldots+\gamma_q-\gamma_{n-p+1}-\ldots-\gamma_n)|u|^2.
&(11.6)\cr}$$
Assume that $\ii\Theta(E)$ is positive. It is then natural to make the
special choice $\omega=\ii\Theta(E)$ for the K\"ahler metric. Then
$\gamma_j=1$ for $j=1,2\ld n$ and we obtain
$\langle[\ii\Theta(E),\Lambda]u,u\rangle=(p+q-n)|u|^2$.
As a consequence:

\begstat{(11.7) Akizuki-Kodaira-Nakano vanishing theorem {\rm ([AN54])}}
If $E$ is a positive line bundle on a compact complex manifold~$X$, then
$$H^{p,q}(X,E)=H^q(X,\Omega^p_X\otimes E)=0\qquad\hbox{for}~~p+q\ge n+1.
\eqno\square$$
\endstat

More generally, if $E$ is a Griffiths positive (or ample) vector bundle
of rank~$r\ge 1$, Le Potier [LP75] proved that $H^{p,q}(X,E)=0$ for
$p+q\ge n+r$. The proof is not a direct consequence of the Bochner
technique. A rather easy proof has been found by M.~Schneider [Sch74],
using the Leray spectral sequence associated to the projectivized bundle
projection $\bbbp(E)\to X$.

\begstat{(11.8) Exercise} \rm It is important for various applications
to obtain vanishing theorems which are also valid in the case of semipositive
line bundles. The easiest case is the following result of Girbau [Gir76]:
let $(X,\omega)$ be compact K\"ahler; assume that $E$ is a line bundle and
that $\ii\Theta(E)\ge 0$ has at least $n-k$ positive eigenvalues at each
point, for some integer~$k\ge 0$; show that $H^{p,q}(X,E)=0$ for $p+q\ge
n+k+1$.\newline
{\it Hint:}\/ use the K\"ahler metric $\omega_\varepsilon=
\ii\Theta(E)+\varepsilon\omega$ with $\varepsilon>0$ small.
\smallskip
A stronger and more natural ``algebraic version'' of this result has been
obtained by Sommese [Som78, ShSo85]: define $E$ to be $k$-ample if some multiple
$mE$ is such that the canonical map
$$\Phi_{|mE|}:X\ssm B_{|mE|}\to\bbbp^{N-1}$$
has at most $k$-dimensional fibres and $\dim B_{|mE|}\le k$. If
$X$ is projective and $E$ is $k$-ample, show that $H^{p,q}(X,E)=0$ for
$p+q\ge n+k+1$.\newline
{\it Hint:}\/ prove the dual result $H^{p,q}(X,E^{-1})=0$ for $p+q\le n-k-1$
by induction on~$k$. First show that $E$ 0-ample $\Rightarrow$ $E$ positive;
then use hyperplane sections $Y\subset X$ to prove the induction step,
thanks to the exact sequences
$$\eqalignno{
&0\lra\Omega^p_X\otimes E^{-1}\otimes\cO(-Y)\lra\Omega^p_X\otimes E^{-1}
\lra\big(\Omega^p_X\otimes E^{-1}\big)_{\restriction Y}\lra 0,&\cr
&0\lra\Omega^{p-1}_Y\otimes E^{-1}_{\restriction Y}\lra
\big(\Omega^p_X\otimes E^{-1}\big)_{\restriction Y}\lra
\Omega^p_Y\otimes E^{-1}_{\restriction Y}\lra 0.&\square\cr}$$
\endstat

\titleb{\S 12.}{$L^2$ Estimates 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 will only  outline the main ideas, referring e.g.\ to [Dem82]
for a detailed exposition of the technical situation considered here.

\begstat{(12.1) Theorem} Let $(X,\omega)$ be a K\"ahler manifold. Here $X$
is not necessarily compact, but we assume that the geodesic distance
$\delta_\omega$ is complete on~$X$. Let $E$ be a hermitian vector bundle of
rank $r$ over $X$, and assume that the curvature operator
$A=A^{p,q}_{E,\omega}=[\ii\Theta(E),\Lambda_\omega]$ is positive
definite everywhere on $\Lambda^{p,q}T^\star_X\otimes E$, $q\ge 1$.
Then for any form $g\in L^2(X,\Lambda^{p,q}T^\star_X\otimes E)$
satisfying $D''g=0$ and $\int_X\langle A^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)$ such that  $D''f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X\langle A^{-1}g,g\rangle\,dV_\omega.$$
\endstat

\begproof{} The assumption that $\delta_\omega$ is complete implies the
existence of cut-off functions $\psi_\nu$ with arbitrarily large compact
support such that $|d\psi_\nu|\le 1$ (take $\psi_\nu$ to be a function of
the distance $x\mapsto \delta_\omega(x_0,x)$, which is an almost
everywhere differentiable 1-Lipschitz function, and regularize if
necessary). From this, it follows that very form $u\in
L^2(X,\Lambda^{p,q}T^\star_X\otimes E)$ such that $D''u\in L^2$ and
$D^{\prime\prime\star}u\in L^2$ in the sense of distribution theory is a
limit of a sequence of smooth forms $u_\nu$ with compact support, in such
a way that $u_\nu\to u$, $D''u_\nu\to D''u$ and $D^{\prime\prime\star}u_\nu
\to D^{\prime\prime\star}u$ in~$L^2$ (just take $u_\nu$ to be a
regularization of $\psi_\nu u$). As a consequence, the basic a priori
inequality (11.3) extends to arbitrary forms $u$ such that
$u,~D''u,D^{\prime\prime\star}u\in L^2$ . Now, consider the Hilbert
space orthogonal decomposition
$$L^2(X,\Lambda^{p,q}T^\star_X\otimes E)=\Ker
D''\oplus({\Ker D''})^\perp,$$
observing that $\Ker D''$ is weakly (hence strongly) closed. Let $v=v_1+v_2$
be the decomposition of a smooth form $v\in\cD^{p,q}(X,E)$ with compact
support according to this decomposition ($v_1$, $v_2$ do not have compact
support in general$\,$!).
Since $(\Ker D'')^\perp\subset\Ker D^{\prime\prime\star}$ by duality
and $g,v_1\in\Ker D''$ by hypothesis, we get
$D^{\prime\prime\star}v_2=0$ and
$$|\langle g,v\rangle|^2=|\langle g,v_1\rangle|^2\le
\int_X\langle A^{-1}g,g\rangle \,dV_\omega
\int_X\langle Av_1,v_1\rangle \,dV_\omega$$
thanks to the Cauchy-Schwarz inequality. The a priori inequality (11.3)
applied to $u=v_1$ yields
$$\int_X\langle Av_1,v_1\rangle \,dV_\omega\le
\|D''v_1\|^2+\|D^{\prime\prime\star}v_1\|^2=\|D^{\prime\prime\star}v_1\|^2=
\|D^{\prime\prime\star}v\|^2.$$
Combining both inequalities, we find
$$|\langle g,v\rangle|^2\le\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)
\|D^{\prime\prime\star}v\|^2$$
for every smooth $(p,q)$-form $v$ with compact support. This shows that we
have a well defined linear form
$$w=D^{\prime\prime\star}v\longmapsto\langle v,g\rangle,\qquad
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)
\supset D^{\prime\prime\star}(\cD^{p,q}(E))\lra\bbbc$$
on the range of $D^{\prime\prime\star}$. This linear form
is continuous in $L^2$ norm and has norm $\le C$ with
$$C=\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)^{1/2}.$$
By the Hahn-Banach theorem, there is an element
$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)$ with $||f||\le C$, such that
$\langle v,g\rangle=\langle D^{\prime\prime\star}v,f\rangle$ for every $v$,
hence $D''f=g$ in the sense of distributions. The inequality $||f||\le C$
is equivalent to the last estimate in the theorem.\qed
\endproof

The above $L^2$ existence theorem can be applied in the fairly general
context of {\it weakly pseudoconvex} manifolds. By this, we mean a complex
manifold $X$ such that there exists a smooth psh exhaustion function
$\psi$ on $X$ ($\psi$ is said to be an exhaustion if for every $c>0$ the
sublevel set $X_c=\psi^{-1}(c)$ is relatively compact, i.e.\ $\psi(z)$
tends to $+\infty$ when $z$ is taken outside larger and larger compact
subsets of~$X$). In particular, every compact complex manifold $X$ is
weakly pseudoconvex (take $\psi=0$), as well as every Stein
manifold, e.g.\ affine algebraic submanifolds of $\bbbc^N$ (take
$\psi(z)=|z|^2$), open balls $X=B(z_0,r)$ $\big($take $\psi(z)=
1/(r-|z-z_0|^2)\big)$, convex open subsets, etc. Now, a basic observation
is that every weakly pseudoconvex K\"ahler manifold $(X,\omega)$ carries a
{\it complete} K\"ahler metric: let $\psi\ge 0$ be a psh exhaustion function
and set
$$\omega_\varepsilon=\omega+\varepsilon\,\ii d'd''\psi^2=
\omega+2\varepsilon(2\ii\psi d'd''\psi+\ii d'\psi\wedge d''\psi).$$
Then $|d\psi|_{\omega_\varepsilon}\le 1/\varepsilon$ and
$|\psi(x)-\psi(y)|\le\varepsilon^{-1}\delta_{\omega_\varepsilon}(x,y)$. 
It follows easily from this estimate that the geodesic balls are relatively
compact, hence $\delta_{\omega_\varepsilon}$ is complete
for every $\varepsilon>0$. Therefore, the $L^2$ existence theorem can be
applied to each K\"ahler metric $\omega_\varepsilon$, and by passing to
the limit it can even be applied to the non necessarily complete
metric~$\omega$. An important special case is the following

\begstat{(12.2) Theorem} Let $(X,\omega)$ be a K\"ahler manifold,
$\dim X=n$. Assume that $X$ is weakly pseudoconvex. Let $E$ be a hermitian
line bundle and let
$$\gamma_1(x)\le\ldots\le\gamma_n(x)$$
be the curvature eigenvalues $($i.e.\ the eigenvalues of $\ii\Theta(E)$
with respect to the metric~$\omega)$ at every point. Assume that the
curvature is positive, i.e.\ $\gamma_1>0$ everywhere. Then for any form
$g\in L^2(X,\Lambda^{n,q}T^\star_X\otimes E)$ satisfying $D''g=0$ and
\hbox{$\int_X\langle(\gamma_1+\ldots+\gamma_q)^{-1}|g|^2\,dV_\omega<+\infty$},
there exists \hbox{$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)$} such that
$D''f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X
(\gamma_1+\ldots+\gamma_q)^{-1}|g|^2\,dV_\omega.$$
\endstat

\begproof{} Indeed, for $p=n$, Formula~11.6 shows that
$$\langle Au,u\rangle\ge(\gamma_1+\ldots+\gamma_q)|u|^2,$$
hence $\langle A^{-1}u,u\rangle\ge(\gamma_1+\ldots+\gamma_q)^{-1}|u|^2$.\qed
\endproof

An important observation is that the above theorem still applies when the
hermitian metric on $E$ is a singular metric with positive curvature in
the sense of currents. In fact, by standard
regularization techniques (convolution of psh functions by smoothing
kernels), the metric can be made smooth and the solutions obtained by
(12.1) or (12.2) for the smooth metrics have limits satisfying the desired
estimates. Especially, we get the following

\begstat{(12.3) Corollary} Let $(X,\omega)$ be a K\"ahler manifold,
$\dim X=n$. Assume that $X$ is weakly pseudoconvex. Let $E$ be a
holomorphic line bundle equipped with a singular metric whose local
weights are denoted $\varphi\in L^1_\loc$. Suppose that
$$\ii\Theta(E)=2\ii d'd''\varphi\ge\varepsilon\omega$$
for some $\varepsilon>0$. Then for any form $g\in L^2(X,\Lambda^{n,q}
T^\star_X\otimes E)$ satisfying $D''g=0$, there exists
$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)$ such that  $D''f=g$ and
$$\int_X|f|^2e^{-2\varphi}\,dV_\omega\le{1\over q\varepsilon}\int_X
|g|^2e^{-2\varphi}\,dV_\omega.\eqno\square$$
\endstat

Here we denoted somewhat incorrectly the metric by $|f|^2e^{-2\varphi}$,
as if the weight $\varphi$ was globally defined on~$X$ (of course, this
is so only if $E$ is globally trivial). We will use this notation anyway,
because it clearly describes the dependence of the $L^2$ norm on the psh
weights.

\titleb{\S 13.}{Nadel and Kawamata-Viehweg vanishing Theorems}
We first introduce the concept of {\it multiplier ideal sheaf},
following A.~Nadel [Nad89]. The main idea actually goes back to the
fundamental works of Bombieri [Bom70] and H.~Skoda [Sko72].

\begstat{(13.1) Definition} Let $\varphi$ be a
psh function on an open subset $\Omega\subset X\,$; to $\varphi$ is
associated the ideal subsheaf $\cI(\varphi)\subset\cO_\Omega$ of germs of
holomorphic functions $f\in\cO_{\Omega,x}$ such that $|f|^2e^{-2\varphi}$
is integrable with respect to the Lebesgue measure in some local
coordinates near~$x$.
\endstat

The zero variety $V(\cI(\varphi))$ is thus the set of points in
a neighborhood of which $e^{-2\varphi}$ is non integrable. Of course,
such points occur only if $\varphi$ has logarithmic poles. This is made
precise as follows.

\begstat{(13.2) Definition} A psh function $\varphi$ is said to have a
logarithmic pole of coefficient $\gamma$ at a point $x\in X$ if the Lelong
number
$$\nu(\varphi,x):=\liminf_{z\to x}{\varphi(z)\over\log|z-x|}$$
is non zero and if $\nu(\varphi,x)=\gamma$.
\endstat

\begstat{(13.3) Lemma {\rm (Skoda [Sko72])}} Let $\varphi$ be a psh function
on an open set $\Omega$ and let $x\in\Omega$.
\smallskip
\item{\rm a)} If $\nu(\varphi,x)<1$, then $e^{-2\varphi}$ is integrable in
a neighborhood of~$x$, in particular $\cI(\varphi)_x=\cO_{\Omega,x}$.
\smallskip
\item{\rm b)} If $\nu(\varphi,x)\ge n+s$ for some integer $s\ge 0$, then
$e^{-2\varphi}\ge C|z-x|^{-2n-2s}$ in a neighborhood of $x$ and
$\cI(\varphi)_x\subset\gm_{\Omega,x}^{s+1}$, where $\gm_{\Omega,x}$ is the
maximal ideal of $\cO_{\Omega,x}$.
\vskip0pt
\endstat

\begproof{} The proof is based on standard estimates of complex potential
theory, see  Skoda [Sko72].
\endproof

\begstat{(13.4) Proposition {\rm([Nad89])}} For any psh function
$\varphi$ on~$\Omega\subset X$, the sheaf
$\cI(\varphi)$ is a coherent sheaf of ideals over $\Omega$.
\endstat

\begproof{} Since the result is local, we may assume that $\Omega$ is the
unit ball in $\bbbc^n$. Let $\cH_\varphi(\Omega)$ be the set of all holomorphic functions $f$ 
on $\Omega$ such that $\int_\Omega|f|^2e^{-2\varphi}\,d\lambda<+\infty$.
By the strong noetherian property of coherent sheaves, the set $\cH_\varphi(\Omega)$
generates a coherent ideal sheaf $\cJ\subset\cO_\Omega$. It is clear that
$\cJ\subset\cI(\varphi)$; in order to prove the equality,
we need only check that $\cJ_x+\cI(\varphi)_x\cap
\gm^{s+1}_{\Omega,x}=\cI(\varphi)_x$ for every integer $s$, in
view of the Krull lemma. Let $f\in\cI(\varphi)_x$ be defined 
in a neighborhood $V$ of $x$ and let $\theta$ be a cut-off function
with support in $V$ such that $\theta=1$ in a neighborhood of $x$.
We solve the equation $d''u=g:=d''(\theta f)$ by means of H\"ormander's
$L^2$ estimates~12.3, where $E$ is the trivial line bundle
$\Omega\times\bbbc$ equipped with the strictly psh weight 
$$\wt\varphi(z)=\varphi(z)+(n+s)\log|z-x|+|z|^2.$$ 
We get a solution $u$ such that
$\int_\Omega |u|^2e^{-2\varphi}|z-x|^{-2(n+s)}d\lambda<\infty$, thus
$F=\theta f-u$ is holomorphic, $F\in \cH_\varphi(\Omega)$ and 
$f_x-F_x=u_x\in\cI(\varphi)_x\cap\gm_{\Omega,x}^{s+1}$.
This proves our contention.\qed
\endproof

The multiplier ideal sheaves satisfy the following basic
fonctoriality property with respect to direct images of sheaves
by modifications.

\begstat{(13.5) Proposition} Let $\mu:X'\to X$ be a modification
of non singular complex manifolds $($i.e.\ a proper generically 1:1
holomorphic map$)$, and let $\varphi$ be a psh function on~$X$. Then 
$$\mu_\star\big(\cO(K_{X'})\otimes\cI(\varphi\circ\mu)\big)=
\cO(K_X)\otimes\cI(\varphi).$$
\endstat

\begproof{} Let $n=\dim X=\dim X'$ and let $S\subset X$ be an analytic
set such that $\mu:X'\ssm S'\to X\ssm S$ is a biholomorphism. By
definition of multiplier ideal sheaves, $\cO(K_X)\otimes\cI(\varphi)$ is
just the sheaf of holomorphic $n$-forms $f$ on open sets $U\subset X$
such that
$\ii^{n^2}f\wedge\ovl f\,e^{-2\varphi}\in L^1_\loc(U)$. Since
$\varphi$ is locally bounded from above, we may even consider forms $f$
which are a priori defined only on $U\ssm S$, because $f$ will be
in $L^2_\loc(U)$ and therefore will automatically extend through~$S$.
The change of variable formula yields
$$\int_U\ii^{n^2}f\wedge\ovl f\,e^{-2\varphi}=
\int_{\mu^{-1}(U)}\ii^{n^2}\mu^\star f\wedge\ovl{\mu^\star f}
\,e^{-2\varphi\circ\mu},$$
hence $f\in\Gamma(U,\cO(K_X)\otimes\cI(\varphi))$ iff
$\mu^\star f\in\Gamma(\mu^{-1}(U),\cO(K_{X'})\otimes\cI(\varphi\circ
\mu))$. Proposition~13.5 is proved.\qed
\endproof

\begstat{(13.6) Remark} \rm If $\varphi$ has algebraic singularities
(according to Definition~10.7), the computation of $\cI(\varphi)$
can be reduced to a purely algebraic problem.

The first observation is that $\cI(\varphi)$ can be computed easily if
$\varphi=\sum\alpha_j\log|g_j|$ where $D_j=g^{-1}_j(0)$ are nonsingular
irreducible divisors with normal crossings. Then $\cI(\varphi)$ is the
sheaf of functions $h$ on open sets $U\subset X$ such that
$$\int_U|h|^2\prod|g_j|^{-2\alpha_j}dV<+\infty.$$
Since locally the $g_j$ can be taken to be coordinate functions from a
local coordinate system $(z_1\ld z_n)$, the condition is that
$h$ is divisible by $\prod g_j^{m_j}$ where $m_j-\alpha_j>-1$ for
each $j$, i.e.\ $m_j\ge\lfloor\alpha_j\rfloor$ (integer part). Hence
$$\cI(\varphi)=\cO(-\lfloor D\rfloor)=\cO(-\sum\lfloor\alpha_j\rfloor D_j)$$
where $\lfloor D\rfloor$ denotes the integral part of the $\bbbq$-divisor
$D=\sum\alpha_jD_j$.

Now, consider the case of general algebraic singularities and suppose
that
$$\varphi\sim{\alpha\over 2}\log\big(|f_1|^2+\cdots+|f_N|^2\big)$$
near the poles. By the remarks after Definition~10.7, we may assume
that the $(f_j)$ are generators of the integrally closed ideal
sheaf $\cJ=\cJ(\varphi/\alpha)$, defined as the sheaf of holomorphic
functions $h$ such that $|h|\le C\exp(\varphi/\alpha)$. In this case,
the computation is made as follows.

First, one computes a smooth modification $\mu:\wt X\to X$ of $X$ such
that $\mu^\star\cJ$ is an invertible sheaf $\cO(-D)$ associated with
a normal crossing divisor $D=\sum\lambda_jD_j$, where $(D_j)$ are the
components of the exceptional divisor of $\wt X$ (take the blow-up $X'$
of $X$ with respect to the ideal $\cJ$ so that the pull-back of $\cJ$
to $X'$ becomes an invertible sheaf $\cO(-D')$, then blow up again by
Hironaka [Hi64] to make $X'$ smooth and $D'$ have normal crossings).
Now, we have $K_{\wt X}=\mu^\star K_X+R$ where $R=\sum\rho_j
D_j$ is the zero divisor of the Jacobian function $J_\mu$ of the blow-up
map. By the direct image formula~13.5, we get
$$\cI(\varphi)=\mu_\star\big(\cO(K_{\wt X}-\mu^\star K_X)\otimes
\cI(\varphi\circ\mu)\big)=\mu_\star\big(\cO(R)\otimes
\cI(\varphi\circ\mu)\big).$$
Now, $(f_j\circ\mu)$ are generators of the ideal $\cO(-D)$, hence
$$\varphi\circ\mu\sim\alpha\sum\lambda_j\log|g_j|$$
where $g_j$ are local generators of $\cO(-D_j)$. We are thus reduced to
computing multiplier ideal sheaves in the case where the poles are
given by a $\bbbq$-divisor with normal crossings $\sum\alpha\lambda_jD_j$.
We obtain $\cI(\varphi\circ\mu)=\cO(-\sum\lfloor\alpha\lambda_j\rfloor D_j)$,
hence
$$\cI(\varphi)=\mu_\star\cO_{\wt X}\big(\sum(\rho_j-\lfloor\alpha\lambda_j
\rfloor)D_j\big).\eqno\square$$
\endstat

\begstat{(13.7) Exercise} \rm Compute the multiplier ideal sheaf
$\cI(\varphi)$ associated with
$\varphi=\log(|z_1|^{\alpha_1}+\ldots+|z_p|^{\alpha_p})$
for arbitrary real numbers $\alpha_j>0$.\newline
{\it Hint:} using Parseval's formula and polar coordinates $z_j=r_j
e^{\ii\theta_j}$, show that the problem is equivalent to determining for
which $p$-tuples $(\beta_1\ld\beta_p)\in\bbbn^p$ the integral
$$\int_{[0,1]^p}{r_1^{2\beta_1}\ldots r_p^{2\beta_p}\,r_1dr_1\ldots r_pdr_p
\over r_1^{2\alpha_1}+\ldots+r_p^{2\alpha_p}}
=\int_{[0,1]^p}{t_1^{(\beta_1+1)/\alpha_1}\ldots t_p^{(\beta_p+1)/\alpha_p}
\over t_1+\ldots+t_p}{dt_1\over t_1}\ldots{dt_p\over t_p}$$
is convergent. Conclude from this that $\cI(\varphi)$ is generated by
the monomials $z_1^{\beta_1}\ldots z_p^{\beta_p}$ such that
$\sum(\beta_p+1)/\alpha_p>1$. (This exercise shows that the analytic
definition of $\cI(\varphi)$ is sometimes also quite convenient for
computations).\qed
\endstat

Let $E$ be a line bundle over $X$ with a singular metric of curvature
current $\Theta(E)$. If $\varphi$ is the weight representing the metric
in an open set $\Omega\subset X$, the ideal sheaf $\cI(\varphi)$ is
independent of the choice of the trivialization and so it is the restriction
to $\Omega$ of a global coherent sheaf on $X$ which we shall still write
$\cI(\varphi)$ by abuse of notation. In this context, we have the following
fundamental vanishing theorem, which is probably one of the most central
results of analytic and algebraic geometry (as we will see later, it
contains the Kawamata-Viehweg vanishing theorem as a special case).

\begstat{(13.8) Nadel vanishing theorem {\rm([Nad89], [Dem93b])}} Let
$(X,\omega)$ be a K\"ahler weakly pseudoconvex manifold, and let $E$ be a
holomorphic line bundle over $X$ equipped with a singular hermitian metric
of weight~$\varphi$. Assume that $\ii\Theta(E)\ge\varepsilon\omega$ for some
continuous positive function $\varepsilon$ on~$X$. Then
$$H^q\big(X,\cO(K_X+E)\otimes\cI(\varphi)\big)=0\qquad
\hbox{for all $q\ge 1$.}$$
\endstat

\begproof{} Let $\cL^q$ be the sheaf of germs of $(n,q)$-forms 
$u$ with values in $E$ and with measurable coefficients, such that
both $|u|^2e^{-2\varphi}$ and $|d''u|^2e^{-2\varphi}$ are locally
integrable. The $d''$ operator defines a complex of sheaves
$(\cL^\bu,d'')$ which is a resolution of the sheaf
$\cO(K_X+E)\otimes\cI(\varphi)$: indeed, the kernel of $d''$ in degree
$0$ consists of all germs of holomorphic $n$-forms with values in $E$
which satisfy the integrability condition; hence the coefficient
function lies in $\cI(\varphi)$; the exactness in degree $q\ge 1$
follows from Corollary~12.3 applied on arbitrary small balls.
Each sheaf $\cL^q$ is a ${\cal C}^\infty$-module, so $\cL^\bu$ 
is a resolution by acyclic sheaves. Let $\psi$ be a smooth psh
exhaustion function on~$X$. Let us apply Corollary~12.3 globally on~$X$,
with the original metric of $E$ multiplied by the factor
$e^{-\chi\circ\psi}$, where $\chi$ is a convex increasing function of
arbitrary fast growth at infinity. This factor can be used to ensure
the convergence of integrals at infinity. By Corollary~12.3, we conclude
that $H^q\big(\Gamma(X,\cL^\bu)\big)=0$ for $q\ge 1$. The theorem
follows.\qed
\endproof 

\begstat{(13.9) Corollary} Let $(X,\omega)$, $E$ and $\varphi$ be as in
Theorem~13.8 and let $x_1\ld x_N$ be isolated points in the zero
variety~$V(\cI(\varphi))$. Then there is a surjective map
$$H^0(X,K_X+E)\lraww\bigoplus_{1\le j\le N}\cO(K_X+L)_{x_j}\otimes
\big(\cO_X/\cI(\varphi)\big)_{x_j}.$$
\endstat

\begproof{} Consider the long exact sequence of cohomology associated to the
short exact sequence $0\to\cI(\varphi)\to\cO_X\to\cO_X/\cI(\varphi)\to 0$
twisted by $\cO(K_X+E)$, and apply Theorem~13.8 to obtain the vanishing of
the first $H^1$ group. The asserted surjectivity property follows.\qed
\endproof

\begstat{(13.10) Corollary} Let $(X,\omega)$, $E$ and $\varphi$ be as in
Theorem~13.8 and suppose that the weight function $\varphi$ is such that
$\nu(\varphi,x)\ge n+s$ at some point $x\in X$ which is an isolated point
of $E_1(\varphi)$. Then $H^0(X,K_X+E)$ generates all $s$-jets at~$x$.
\endstat

\begproof{} The assumption is that $\nu(\varphi,y)<1$ for $y$ near $x$,
$y\ne x$. By Skoda's lemma 13.3~b), we conclude that $e^{-2\varphi}$ is
integrable at all such points~$y$, hence $\cI(\varphi)_y=\cO_{X,y}$, whilst
$\cI(\varphi)_x\subset\gm_{X,x}^{s+1}$ by 13.3~a). Corollary 13.10 is thus
a special case of 13.9.\qed
\endproof

The philosophy of these results (which can be seen as generalizations
of the H\"ormander-Bombieri-Skoda theorem [Bom70], [Sko72,~75]) is that
the problem of constructing holomorphic sections of $K_X+E$ can be
solved by constructing suitable hermitian metrics on $E$ such that the
weight $\varphi$ has isolated poles at given points~$x_j$.

\begstat{(13.11) Exercise} \rm Assume that $X$ is compact and that $L$ is
a positive line bundle on~$X$. Let $\{x_1\ld x_N\}$ be a finite set.
Show that there are constants $a,b\ge 0$ depending only on $L$ and $N$
such that $H^0(X,mL)$ generates jets of any order $s$ at all points $x_j$
for $m\ge as+b$.\newline
{\it Hint:} Apply Corollary~13.9 to $E=-K_X+mL$, with a singular
metric on $L$ of the form $h=h_0e^{-\varepsilon\psi}$, where $h_0$ is
smooth of positive curvature, $\varepsilon>0$ small and
$\psi(z)\sim\log|z-x_j|$ in a neighborhood of $x_j$.\newline
Derive the Kodaira embedding theorem from the above result:
\endstat

\begstat{(13.12) 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.\qed
\endstat

An equivalent way of stating the Kodaira embedding theorem is:

\begstat{(13.13) Kodaira projectivity criterion} A compact complex manifold
$X$ is projective algebraic if and only $X$ carries a Hodge metric, that is,
a K\"ahler metric with integral cohomology class.
\endstat

\begproof{} If $X\subset\bbbp^N$ is projective algebraic, then the
restriction of the Fubini-Study metric to $X$ is a Hodge metric.
Conversely, if $X$ has a Hodge metric, then any representative of
$\{\omega\}$ in $H^2(X,\bbbz)$ defines a $\ci$ complex line bundle~$L$.
From the fact that $\omega$ is of type $(1,1)$ and from exponential
exact sequence (8.20)
$$H^1(X,\cO_X^\star)\to H^2(X,\bbbz)\to H^2(X,\bbbc)=H^{0,2}(X,\bbbc)$$
we infer that we can represent this line bundle by a cocycle in
$H^1(X,\cO_X^\star)$, in other words, $L$ can be given a holomorphic
structure. Moreover, there is a hermitian metric $h$ on $L$ such that
${\ii\over 2\pi}\Theta_h(L)=\omega$. Therefore $L$ is ample
and $X$ is projective.\qed
\endproof

\begstat{(13.14) Exercise {\rm (Riemann condition for abelian varieties)}}
\rm A complex torus $X=\bbbc^n/\Gamma$ is called an {\it Abelian variety}
if $X$ is projective algebraic. Show by using (13.13) that this is the case
if and only if there exists a positive definite hermitian form
$H$ on $\bbbc^n$ such that $\Im H(\gamma_1,\gamma_2)\in\bbbz$ for all
$\gamma_1,\gamma_2$ in the lattice $\Gamma$.\hfill\break
{\it Hint\/}: use an averaging process to reduce the proof to the case
of K\"ahler metrics which are invariant by translations.
\endstat

\begstat{(13.15) Exercise {\rm (solution of the Levi problem)}} \rm Show
that the following two properties are equivalent.
\smallskip
\item{\rm a)} $X$ is strongly pseudoconvex, i.e.\ $X$ admits a strongly psh
exhaustion function.
\smallskip
\item{\rm b)} $X$ is Stein, i.e.\ the global holomorphic functions
$H^0(X,\cO_X)$ separate points and yield local coordinates at any point,
and $X$ is holomorphically convex $($this means that for any discrete
sequence $z_\nu$ there is a function $f\in H^0(X,\cO_X)$ such that
$|f(z_\nu)|\to\infty)$.\qed
\vskip0pt
\endstat

\begstat{(13.16) Remark} \rm As long as forms of bidegree $(n,q)$ are
considered, the $L^2$ estimates can be extended to complex spaces with
arbitrary singularities. In fact, if $X$ is a complex space and $\varphi$
is a psh weight function on~$X$, we may still define a sheaf $K_X(\varphi)$
on~$X$, such that the sections on an open set $U$ are the holomorphic
$n$-forms $f$ on the regular part $U\cap X_\reg$, satisfying the
integrability condition $\ii^{n^2}f\wedge\ovl f\,e^{-2\varphi}\in
L^1_\loc(U)$. In this setting, the fonctoriality property 13.5 becomes
$$\mu_\star\big(K_{X'}(\varphi\circ\mu)\big)=K_X(\varphi)$$
for arbitrary complex spaces $X$, $X'$ such that $\mu:X'\to X$
is a modification. If $X$ is nonsingular we have $K_X(\varphi)=
\cO(K_X)\otimes\cI(\varphi)$, however, if $X$ is
singular, the symbols $K_X$ and $\cI(\varphi)$ must not be
dissociated. The statement of the Nadel vanishing theorem becomes
$H^q(X,\cO(E)\otimes K_X(\varphi))=0$ for $q\ge 1$, under the same
assumptions ($X$ K\"ahler and weakly pseudoconvex, curvature
$\ge\varepsilon\omega$). The proof can be obtained by restricting
everything to $X_\reg$. Although in general $X_\reg$ is not weakly
pseudoconvex (e.g.\ in case $\codim X_\sing\ge 2$), $X_\reg$ is
always K\"ahler complete (the complement of a proper analytic subset in
a K\"ahler weakly pseudoconvex space is complete K\"ahler, see
e.g.\ [Dem82]). As a consequence, Nadel's vanishing theorem is
essentially insensitive to the presence of singularities.\qed
\endstat

We now derive an algebraic version of the Nadel vanishing theorem
obtained independently by Kawamata [Kaw82] and Viehweg [Vie82] (with a
completely different method based on the use of cyclic coverings).
Before stating the theorem, we need a definition.

\begstat{(13.17) Definition} A line bundle $L$ on a projective algebraic
variety is said to be numeri\-cally effective $($nef for short$)$ if $L$
satisfies the following two equivalent properties:
\smallskip
\item{\rm i)} For every irreducible algebraic curve $C\subset X$, one has
$L\cdot C=\int_C c_1(L)\ge 0$.
\smallskip
\item{\rm ii)} If $A$ is an ample line bundle, then $kL+A$ is ample
for all $k\ge 0$.
\smallskip\noindent
The line bundle $L$ is said to be big if its Kodaira dimension is equal
to $n=\dim X$, that is, if there exists a constant $c>0$ such that
$$H^0(X,kL)\ge c\,k^n,\qquad k\ge k_0.$$
\endstat

The equivalence of Properties 13.17 i) and ii) is well known and
will be taken for granted (a proof can be found in Hartshorne [Har70]).
Now, if $D=\sum\alpha_jD_j\ge 0$ is an effective $\bbbq$-divisor,
we define the {\it multiplier ideal sheaf} $\cI(D)$ to be equal to
$\cI(\varphi)$ where $\varphi=\sum\alpha_j|g_j|$ is the corresponding psh
function defined by generators $g_j$ of $\cO(-D_j)\,$; as we saw in
Remark~13.6, the computation of $\cI(D)$ can be made algebraically by
using desingularizations \hbox{$\mu:\wt X\to X$} such that $\mu^\star D$
becomes a divisor with normal crossings on~$\wt X$.

\begstat{(13.18) Kawamata-Viehweg vanishing theorem} Let $X$ be a projective
algebraic mani\-fold and let $F$ be a line bundle over $X$ such that some
positive multiple $mF$ can be written $mF=L+D$ where $L$ is a 
big nef line bundle and $D$ an effective divisor. Then
$$H^q\big(X,\cO(K_X+F)\otimes\cI(m^{-1}D)\big)=0~~~~\hbox{\it for}~~q\ge 1.$$
\endstat

\begstat{(13.19) Corollary} If $F$ is big and nef, then
$H^q\big(X,\cO(K_X+F))=0$ for $q\ge 1$.
\endstat

\begproof{} Let $A$ be a non singular very ample divisor. We have an exact
sequence
$$0\to H^0(X,\cO(kL-A))\to H^0(X,\cO(kL))\to H^0(A,\cO(kL)_{\restriction A})$$
in which $\dim H^0(A,\cO(kL)_{\restriction A})\le C\,k^{n-1}$ for some
constant $C\ge 0$. As $L$ is big, there is an integer $k_0\gg 0$ such that
$\cO(k_0L-A)$ has a non trivial section. If $E$ is the divisor of this
section, we find $\cO(k_0L-A)\simeq\cO(E)$, hence
$\cO(k_0L)\simeq\cO(A+E)$. Now, for $k\ge k_0$, we find
$\cO(kL)=\cO((k-k_0)L+A+E)$. By 13.17~ii), the line bundle $\cO((k-k_0)L+A)$
is ample, so it can be equipped with a smooth hermitian metric
$h_k=e^{-\varphi_k}$ of positive definite curvature form
$\omega_k={\ii\over 2\pi}\Theta((k-k_0)L+A)$. Let
$\varphi_D=\sum\alpha_j\log|g_j|$ be the weight of the singular metric on
$\cO(D)$ described in Example~10.20, such that
${\ii\over 2\pi}\Theta(\cO(D))=[D]$, and similarly, let $\varphi_E$ be
such that ${\ii\over 2\pi}\Theta(\cO(E))=[E]$.
We define a singular metric on $\cO(kL)=\cO((k-k_0)L+A+E)$ by the
weight $\varphi_k+\varphi_E$, and we then get a singular metric on
$\cO(mF)=\cO(L+D)$ by taking ${1\over k}(\varphi_k+\varphi_E)+\varphi_D$.
Finally, we get a metric on $F$ of weight
$$\varphi_F={1\over km}(\varphi_k+\varphi_E)+{1\over m}\varphi_D.$$
The corresponding curvature form is
$${\ii\over 2\pi}\Theta(F)={1\over km}(\omega_k+[E])+{1\over m}[D]\ge
{1\over km}\omega_k>0.$$
Moreover $\varphi_F$ has algebraic singularities, and by taking $k$
large enough we find
$$\cI(\varphi_F)=\cI\Big({1\over km}E+{1\over m}D\Big)=
\cI\Big({1\over m}D\Big).$$
In fact, $\cI(\varphi_F)$ can be computed by taking integer parts of
$\bbbq$-divisors (as explained in Remark~13.6), and adding
${1\over km}E+{1\over m}D$ does not change the contribution of ${1\over m}D$
when $k$ is large. Nadel's theorem thus
implies the desired vanishing result for all $q\ge 1$.\qed
\endproof

\titleb{\S 14.}{Algebraic Approach to the Fujita Conjecture}
Given an ample line bundle $L$, a basic question is to compute an
effective integer $m_0$ such that $mL$ is very ample for $m\ge m_0$.
The example of a curve of genus $g$ shows that $m_0$ cannot be less
than $g$, so $m_0$ cannot be a universal value. However, if $mL$ is
replaced by the adjoint line bundle $K_X+mL$, a simple uniform answer
is expected to emerge.

\begstat{(14.1) Fujita conjecture {\rm([Fuj87])}} If $L$ is an ample line
bundle on a projective $n$-dimensional manifold, then
\smallskip
\item{\rm i)} $K_X+(n+1)L$ is generated by global sections$\,;$
\smallskip
\item{\rm ii)} $K_X+(n+2)L$ is very ample.
\vskip0pt
\endstat

The bounds predicted by the conjecture cannot be improved, since
$K_X=\cO(-n-1)$ for the complex projective space $X=\bbbp^n$
(we take of course $L=\cO(1)$). The conjecture is easy to prove
for curves, and I.~Reider [Rei88] settled the conjecture in the
affirmative for $n=2$. So far, the conjecture remains unsettled for
$n\ge 3$ (Ein-Lazarsfeld [EL93] and Fujita [Fuj93, Fuj94] came
close to solving the $3$-dimensional case, part i) is now solved for
$n=3$). Other important results in this direction have been obtained
by Koll\'ar [Kol92]; we refer to [Laz93] for an excellent review
of these developments. In 1993, Siu [Siu93] obtained an explicit value
of an integer $m_0$ such that $m_0L$ becomes very ample, depending only
on the Chern numbers $L^n$, $K_X\cdot L^{n-1}$ (effective version of
the ``big Matsusaka theorem'', see [Mat72] and [KoM83]); improved
estimates can be found in [Dem94].

This section is devoted to a proof of various results related to the
Fujita conjecture in arbitrary dimension. The main ideas occuring here
are inspired by a very recent work of Y.T.~Siu [Siu94]. His method, which
is algebraic in nature and quite elementary, consists in a combination of the
Riemann-Roch formula together with the Kawamata-Viehweg vanishing theorem
(however, it will be much more convenient to use it in Nadel's version
using multiplier ideal sheaves). In the sequel, $X$ denotes a
projective algebraic $n$-dimensional manifold. The first observation is
the following well-known consequence of the Riemann-Roch formula:

\begstat{(14.2) Special case of Riemann-Roch} Let $\cJ\subset\cO_X$ be a
coherent ideal sheaf on $X$ such that the zero variety $V(\cJ)$ has
dimension $d$ $($with possibly some lower dimensional components$)$.
Let $Y=\sum\lambda_jY_j$ be the effective algebraic cycle of dimension
$d$ associated to the $d$ dimensional components of~$V(\cJ)$ $($taking
into account multiplicities $\lambda_j$ given by the ideal $\cJ)$. Then
for any line bundle $E$, the Euler characteristic $\chi(X,\cO(E+mL))$
is a polynomial $P(m)$ of degree $d$ and leading coefficient
$L^d\cdot Y/d!$
\endstat

The second fact is a simple lemma about numerical polynomials (polynomials
with rational coefficients, mapping $\bbbz$ into $\bbbz$).

\begstat{(14.3) Lemma} Let $P(m)$ be a numerical polynomial of degree $d>0$
and leading coefficient $a_d/d!$, $a_d\in\bbbz$, $a_d>0$. Suppose that
$P(m)\ge 0$ for $m\ge m_0$. Then
\smallskip
\item{\rm a)} For every integer $N\ge0$, there exists $m\in[m_0,m_0+Nd]$
such that $P(m)\ge N$.
\smallskip
\item{\rm b)} For every $k\in\bbbn$, there exists $m\in[m_0,m_0+kd]$ such
that \hbox{$P(m)\ge a_dk^d/2^{d-1}$}.
\smallskip
\item{\rm c)} For every $N\ge 3d^2$, there exists
$m\in[m_0,m_0+1,\ldots,m_0+N]$ such that $P(m)\ge N$.
\vskip0pt
\endstat

\begproof{} a) Each of the $N$ equations $P(m)=0$, $P(m)=1$,
$\ldots$, $P(m)=N-1$ has at most $d$ roots, so there must be an integer
$m\in[m_0,m_0+dN]$ which is not a root of these.
\smallskip\noindent
b) By Newton's formula for iterated differences $\Delta P(m)=P(m+1)-P(m)$,
we get
$$\Delta^dP(m)=\sum_{1\le j\le d}(-1)^j{d\choose j}P(m+d-j)=a_d,
\qquad\forall m\in\bbbz.$$
Hence if $j\in\big\{0,2,4\ld 2\lfloor d/2\rfloor\big\}\subset[0,d]$ is the
even integer achieving the maximum of $P(m_0+d-j)$ over this finite set,
we find
$$2^{d-1}P(m_0+d-j)=\left({d\choose 0}+{d\choose 2}+\ldots\right)P(m_0+d-j)
\ge a_d,$$
whence the existence of an integer $m\in[m_0,m_0+d]$ with $P(m)\ge
a_d/2^{d-1}$. The case $k=1$ is thus proved. In general, we apply the
above case to the polynomial $Q(m)=P(km-(k-1)m_0)$, which has leading
coefficient $a_dk^d/d!$
\smallskip\noindent
c) If $d=1$, a) already yields the result. If $d\ge 2$, apply b) with $k$
equal to the smallest integer $k$ such that $k^d/2^{d-1}\ge
N$, i.e.\ $k=\lceil 2^{1-1/d}N^{1/d}\rceil$. Then
$kd\le d(2^{1-1/d}N^{1/d}+1)\le N$ whenever $N\ge 3d^2$, as a short
computation shows.\qed
\endproof

We now use Nadel's vanishing theorem to prove the following results.
The first is due to Fujita [Fuj87] (with a more involved proof based on
Mori theory), and the second is an improvement of a result obtained
recently by Siu [Siu94].

\begstat{(14.4) Theorem {\rm ([Fuj87])}} If $L$ is an ample line bundle over
a projective manifold $X$ of dimension~$n$, then
\smallskip
\item{\rm a)} $K_X+(n+1)L$ is nef$\,;$
\smallskip
\item{\rm b)} $K_X+(n+2)L$ is ample.
\vskip0pt
\endstat

\begstat{(14.5) Theorem} Let $L$ be an ample line bundle on a projective
$n$-fold~$X$. Let $x_1\ld x_p$ be arbitrary points in $X$ and
let $s_1\ld s_p\in\bbbn$ be given integers. For
$$m\ge 2+\sum_{1\le j\le p}{3n+2s_j-1\choose n}$$
there is a surjective map
$$H^0(X,2K_X+mL)\lraww\bigoplus_{1\le j\le N}\cO(2K_X+mL)\otimes\cO_{X,x_j}/
\gm_{X,x_j}^{s_j+1},$$
i.e.\ $2K_X+mL$ generates simultaneously jets of order $s_j$ at all
points~$x_j$. In~particular $2K_X+mL$ is very ample for
$m\ge 2+{3n+1\choose n}$.
\endstat

\begstat{(14.6) Theorem} Let $L$ be an ample line bundle on a
projective $n$-fold~$X$, and let $s\ge 0$, $N\ge 1$ be given integers.
Then for every subset \hbox{$\{x_1\ld x_N\}\subset X$}, the line bundle
$$(2n+2s+1)K_X+2(n+s)(n(N+1)+1)L$$
generates simultaneously jets of order $s$ at all points~$x_j$.
In particular, for $s=1$ and $N=2$, we find that
$$(2n+3)K_X+(2n+2)(3n+1)L$$
is very ample.
\endstat

\begproof{of 14.4, 14.5, 14.6.} We first show that 14.4~a) is a formal
consequence of 14.3~b). In fact, suppose that 14.4~b) is proved and
that we know a value $c_0\in\bbbq^+$ such that
$$\forall L~\hbox{ample},\quad K_X+c_0L~\hbox{is ample as a
$\bbbq$-divisor}.\leqno(\star)$$
By 14.4~b), $c_0=n+2$ is such a value. Then, applying $(\star)$ to
$L'=K_X+(n+2)L$ which is ample, we get
$$K_X+c_0L'=(c_0+1)K_X+(n+2)c_0L~\hbox{ample}\Longrightarrow
K_X+c_1L~\hbox{ample}$$
where $c_1=(n+2)c_0/(c_0+1)$. Iterating this procedure by the recursion
formula $c_{\nu+1}=(n+2)c_\nu/(c_\nu+1)$, we see that $K_X+c_\nu L$ is ample
for all~$\nu$. However, it is easy to see that $\lim c_\nu=n+1$,
hence $K_X+(n+1)L$ is nef.
\smallskip
For the proof of 14.5, we use a more sophisticated recursion procedure. We
start with an integer $m_0\gg 0$ such that $K_X+mL$ is ample for $m\ge m_0$,
where $m_0$ is a priori completely unknown. We then construct inductively a
sequence of singular hermitian metrics $(h_\nu)_{\nu\ge 1}$ on $K_X+m_1L$
with a suitable integer $m_1\le m_0$, such that the weights $\varphi_\nu$
satisfy the following properties:
\smallskip
\item{$\alpha)$\kern-2pt} $\varphi_\nu$ is psh with algebraic singularities;
\smallskip
\item{$\beta)$\kern-2pt} ${\ii\over2\pi}\Theta_{h_\nu}(K_X+m_1 L)
=dd^c\varphi_\nu\ge\varepsilon_\nu\omega$ for some $\varepsilon_\nu>0$ and
some K\"ahler metric $\omega\,$;
\smallskip
\item{$\gamma)$} $\nu(\varphi_\nu,x_j)\ge n+s_j$ for all $j$;
\smallskip
\item{$\delta)$} If $\cI(\varphi_\nu)$ are the multiplier ideal sheaves, then
$\cI(\varphi_{\nu+1})\supset\cI(\varphi_\nu)$, and 
$\cI(\varphi_{\nu+1})\ne\cI(\varphi_\nu)$ whenever the zero variety
$V(\cI(\varphi_\nu))$ has positive dimension.
\smallskip\noindent
Recall that any increasing sequence of coherent subsheaves of a coherent
sheaf is stationary on every compact set (``strong noetherian property''
of coherent sheaves). Hence condition $\delta)$ ensures that the procedure
will finally stop with a zero dimensional subscheme $V(\cI(\varphi_\nu))$.
Observe that once a metric satisfying $(\alpha{-}\delta)$ on $K_X+m_1L$
exists, metrics on $K_X+mL$ with the same properties can be obtained for
$m\ge m_1$ by considering the weight $\varphi_\nu+(m-m_1)\varphi_L$ where
$\varphi_L$ is a weight for a smooth metric with positive definite curvature
on~$L\,$; adding $\varphi_L$ does not change the multiplier ideal sheaf
$\cI(\varphi_\nu)$. Now suppose that $\varphi_\nu$ has been constructed.
By Nadel's vanishing theorem applied to $L'_m=K_X+mL$, $m\ge m_1$, we get
$$H^q(X,\cO(2K_X+mL)\otimes\cI(\varphi_\nu))=0\quad
\hbox{\rm for $q\ge 1$, $m\ge m_1$.}$$
On the other hand, the Kodaira vanishing theorem applied to 
the ample line bundle $L'_m$, $m\ge m_0$, yields
$$H^q(X,\cO(2K_X+mL))=0\quad\hbox{\rm for $q\ge 1$, $m\ge m_0$.}$$
Let us consider the exact sequence $0\to\cI(\varphi_\nu)\to\cO_X\to
\cO_X/\cI(\varphi_\nu)\to 0$ twisted by the invertible sheaf $\cO(2K_X+mL)$.
The above vanishing results imply
$$H^q(X,\cO(2K_X+mL)\otimes\cO_X/\cI(\varphi_\nu))=0\quad
\hbox{\rm for $q\ge 1$, $m\ge m_0$.}$$
For $m\ge m_0$, we apply Lemma 14.3 to the numerical polynomial
$$\eqalign{P(m)
&=\chi(X,\cO(2K_X+mL)\otimes\cO_X/\cI(\varphi_\nu))\cr
&=h^0(X,\cO(2K_X+mL)\otimes\cO_X/\cI(\varphi_\nu))\ge 0,\cr}$$
in order to get a non trivial section
$$\sigma\in H^0(X,\cO(2K_X+mL)\otimes\cO_X/\cI(\varphi_\nu)),\qquad
m\ge m_0.$$
Observe that any such section, viewed as a section of $2K_X+mL$ on the
subscheme $V(\cI(\varphi_\nu))$, has an extension $\wt\sigma$ to $X$ since
$$H^1(X,\cO(2K_X+mL)\otimes\cI(\varphi_\nu))=0,\qquad m\ge m_1.$$
Fix a basis $\sigma_1\ld\sigma_N$ of the sections on $V(\cI(\varphi_\nu))$
and take arbitrary extensions $\wt\sigma_1\ld\wt\sigma_N$ to $X$.
Look at the linear map assigning the collection of jets of order $2n+2s_j-1$
at all points $x_j$
$$\wt\sigma=\sum_{1\le j\le N}a_j\wt\sigma_j\longmapsto
\bigoplus J^{2n+2s_j-1}_{x_j}(\wt\sigma).$$
Since the rank of the bundle of $s$-jets is ${n+s\choose n}$, the target
space has dimension
$$\delta=\sum_{1\le j\le p}{3n+2s_j-1\choose n}.$$
If $\{x_j\}\ne\emptyset$, we have $\delta+1\ge{3n-1\choose n}+1\ge 3n^2$
for $n\ge 3$ (in case \hbox{$n=2$}, we have $\delta+1\ge 3.2^2$ except
when $p=1$ and $s_1=0$, but Theorem 14.5 is anyway true in that case, as
follows from Reider's theorem below). Lemma 14.3~c) gives us an integer
\hbox{$m'\in[m_0,m_0+\delta+1]$} such that \hbox{$N=P(m')\ge\delta+1$}.
Then there exists a section $\wt\sigma\in H^0(X,2K_X+m'L)$ with non trivial
restriction $\sigma$ to $V(\cI(\varphi_\nu))$, which vanishes at
order $\ge 2n+2s_j$ at each point $x_j$. We may always suppose that
our initial integer $m_0$ is chosen such that $m_0\ge\delta+2$ and we then
fix the choice of $m_1$ to be
$$m_1=\left\lceil{m_0+\delta+2\over 2}\right\rceil>{m'\over 2}.$$
We now set $\chi(x,y)={1\over4}\log(e^{4x}+e^{4y})$ and
$$\varphi_{\nu+1}=\chi\left(\varphi_\nu~,~
\log|\wt\sigma|^{1/2}+\left(m_1-{m'\over2}\right)\varphi_L\right).$$
This defines a metric satisfying all properties $(\alpha{-}\delta)$.
In fact, $\alpha,\beta)$ are obvious, because the composition of two
strictly psh functions with $\chi$ is strictly psh (for this, use the fact
that $\chi$ has the ``homogeneity'' property $\chi(x+t,y+t)=\chi(x,y)+t$).
Property $\gamma)$ is true since $\log|\wt\sigma|^{1/2}$ has a logarithmic
pole of coefficient at least $n+s_j$ at $x_j$. The inclusion $\cI(
\varphi_{\nu+1})\supset\cI(\varphi_\nu)$ is immediate from the fact that
$\varphi_{\nu+1}\ge\varphi_\nu$. By construction, $\cI(\varphi_{\nu+1})$
contains all germs of holomorphic functions whose zero divisor contains
the zero divisor of $\sigma$ (or even just $1/2$ that of $\sigma$). As
$\sigma$ does not vanish identically on $V(\cI(\varphi_\nu))$, this
implies that $\cI(\varphi_{\nu+1})\supsetneq\cI(\varphi_\nu)$, whence
property~$\delta)$.

We still have to define the weight $\varphi_1$ in order to initialize the
procedure. We then simply apply Lemma~14.3 to
$$P(m)=h^0(X,\cO(2K_X+mL)),\quad m\ge m_0,$$
so as to get a non trivial section
$\wt\sigma\in H^0(X,\cO(2K_X+m'L))$, $m'\le m_0+\delta+1$, such that
$\wt\sigma$ vanishes at order $\ge 2n+2s_j$ at all points $x_j$. We then set
$$\varphi_1=\log|\wt\sigma|^{1/2}+\left(m_1-{m'\over 2}\right)
\varphi_L.$$
For the proof of 14.4~b), the set $\{x_j\}$ is taken to be empty; the
only thing that we need is to find at each step a section $\wt\sigma$
with non trivial restriction $\sigma$ to $V(\cI(\varphi_\nu))$. In this
case, it is sufficient to have $P(m')\ge1$, hence by 14.3~a), we need
only take $m'\le m_0+n$, and thus $m_1=\lceil(m_0+n+1)/2\rceil$.
Since the final weight $\varphi_\nu$ only has isolated poles, we can
even remove the poles by taking the maximum with a function of the
form $|z-z_0|^2-C$ in a neighborhood of each pole $z_0$, so as to get
a smooth metric with positive definite curvature (for this we use a
regularized max function $\max_\varepsilon=\max(\bu,\bu)\star
\rho_\varepsilon$). This shows that $K_X+m_1L$ is ample. We can iterate
this procedure and obtain inductively that $K_X+m_\nu L$ is ample with
a sequence $m_\nu$ satisfying the recursion formula $m_{\nu+1}=
\lceil(m_\nu+n+1)/2\rceil$. The procedure stops at $m_\nu=n+2$ after a
finite number of steps. Property 14.4~b) is proved.

For the proof of 14.5, $\delta+2$ is much larger than $n+2$, and we can thus
start with $m_0=\delta+2$ (since 14.4~b) is now proved). We then find
$$m_1=m_0=\delta+2=2+\sum{3n+2s_j-1\choose n}$$
and get metrics $\varphi$ on $L'_m=K_X+mL$ satisfying $(\alpha{-}\delta)$
for $m\ge m_0$. An application of Corollary~12.13 to $L'_m$ concludes the
proof.

For the proof of Theorem 14.6, we replace condition $\gamma)$ by the
weaker condition $\nu(\varphi_\nu,x_j)\ge{1\over2}$ for $j=1,\ld N$.
At each step, we thus want to find sections $\wt\sigma$ vanishing at
$x_1\ld x_N$ but not identically on the subscheme
$V(\cI(\varphi_\nu))$. This requires having $P(m')\ge N+1$, which is
the case for some $m'\in[m_0,m_0+n(N+1)]$ by 14.3~a). We can then take
$m_1=m_0=n(N+1)+1>m'/2$. From the existence of the hermitian metric on
$L'=K_X+m_1L$, we conclude again by Corollary 12.13 that
$$K_X+2(n+s)L'=(2n+2s+1)K_X+2(n+s)(n(N+1)+1)L$$
generates simultaneously $s$-jets at all points $x_j$.\qed
\endproof

\begstat{(14.7) Remark} \rm The proof actually shows much more that the
conclusions stated in 14.5 and 14.6: for example, under the numerical condition
$m\ge 2+\sum{3n+2s_j-1\choose n}$, the bundle $K_X+mL$ admits a hermitian
metric with positive definite curvature $\ge\varepsilon\omega$ and with
isolated poles of Lelong number $\ge n+s_j$ at each point $x_j$ in a given
subset. This means that we have obtained in fact a lower bound for the
Seshadri constants $\varepsilon(K_X+mL,x)$. If we take $N=1$, the proof
of Theorem 14.6 tells us that $\varepsilon(K_X+(2n+1)L,x)>1/2$ at any
point.\qed
\endstat

We now list a few immediate consequences of Theorem~14.5, in
connection with some classical questions of algebraic geometry.

\begstat{(14.8) Corollary} Let $X$ be a projective $n$-fold of
general type with $K_X$ ample. Then $mK_X$ is very ample for $m\ge
m_0=6n^2+10n+5$.
\endstat

\begstat{(14.9) Corollary} Let $X$ be a Fano $n$-fold, that is,
a $n$-fold such that $-K_X$ is ample. Then $-mK_X$ is very ample for
$m\ge m_0=6n^2+6n-1$.
\endstat

\begproof{} Corollaries 14.8, 14.9 follow easily from Theorem 14.6 applied
to \hbox{$L=\pm K_X$}. Hence we get pluricanonical embeddings
$\Phi:X\to\bbbp^N$ such that
$\Phi^\star\cO(1)=\pm m_0K_X$. The image $Y=\Phi(X)$ has
degree
$$\deg(Y)=\int_Y c_1\big(\cO(1)\big)^n=
\int_X c_1\big(\pm m_0K_X\big)^n=m_0^n|K_X^n|.$$
It can be easily reproved from this that there are only finitely
many deformation types of Fano $n$-folds, as well as of $n$-folds
of general type with $K_X$ ample, corresponding to a given
discriminant $|K_X^n|$ (from a theoretical viewpoint, this result is
a consequence of Matsusaka's big theorem [Mat72] and [KoM72], but
the bounds which can be obtained from it are probably extremely huge).
In the Fano case, a fundamental result obtained independently by
Koll\'ar-Miyaoka-Mori [KoMM92] and Campana [Cam92] shows that the
discriminant $K_X^n$ is in fact bounded by a constant $C_n$ depending
only on~$n$. Therefore, one can find an explicit bound $C'_n$ for the
degree of the embedding~$\Phi$, and it follows that there are only
finitely many families of Fano manifolds in each dimension.\qed
\endproof

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\eightpoint\parskip=2pt plus 2pt
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d'un fibr\'e vectoriel holomorphe semi-positif au dessus d'une vari\'et\'e
k\"ahl\'e\-rienne compl\`ete|Ann.\ Sci.\ Ec.\ Norm.\ Sup.|15|1982|457-511|

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Proceedings of the Conference ``Geometrical and algebraical aspects
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June {\oldstyle 1989}|

\miscell Dem90|J.-P.\ Demailly|Singular hermitian metrics on positive
line bundles|Proc.\ Conf.\ Complex algebraic varieties (Bayreuth,
April~2-6, 1990), edited by K.~Hulek, T.~Peternell, M.~Schneider,
F.~Schreyer, Lecture Notes in Math., Vol.~1507, Springer-Verlag, Berlin,
{\oldstyle 1992}|

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bundles|J.\ Differential Geom.|37|1993|323-374|

\miscell Dem94|J.-P.\ Demailly|$L^2$ vanishing theorems and adjunction
theory|Lectures Notes of a CIME course on ``Transcendental methods of
algebraic geometry'', ed.\ F.~Catanese and C.~Ciliberto, Cetraro,
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\vskip30pt
\noindent(November 24, 1994)
\vskip10pt
\noindent
Jean-Pierre Demailly\hfill\break
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, URA 188 du CNRS, BP 74,\hfill\break
F-38402 Saint-Martin d'H\`eres, France

\noindent{\it e-mail address:}\/ demailly@fourier.grenet.fr
\end



