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MR0000054 (83g:32007) 32C25 32C30 Barlet, D. Majoration du volume des fibres génériques et forme géométrique du théorème d’aplatissement. (French) [Upper bound of the volume of the generic fibers and a geometric form of the flattening theorem] Séminaire Pierre Lelong-Henri Skoda (Analyse). Années 1978/79 (French), pp. 1–17, Lecture Notes in Math., 822, Springer, Berlin-New York, 1980. The author gives a new proof of the following fact which does not use Hironaka’s flattening theorem: Let π: X → S be a proper surjective morphism between irreducible reduced analytic spaces with n = dim X − dim S. Then there is a modification λ: S̃ → S with the following properties. Let X̃ be the irreducible component of the fiber product X ×_{S} S̃ which is mapped surjectively onto X. Then the natural morphism π̃: S̃ → S̃ is equidimensional. Moreover, equipped with appropriate multiplicities, the fibers {π̃^{−1}(t)}_{t∈S̃} form a family of compact analytic cycles of pure dimension n of X̃ parametrized by S̃.

Takao Fujita (Tokyo)

MR0592787 (82f:14007) 14C05 32F10 32G10 Barlet, D. Convexité au voisinage d’un cycle. (French) Functions of several complex variables, IV (Sem. François Norguet, 1977–1979) (French), pp. 102–121, Lecture Notes in Math., 807, Springer, Berlin, 1980. The author obtains a natural local coordinatization of the Chow scheme of n cycles of a reduced (and smooth or quasiprojective) variety Z by utilizing integration of elements of H^{n}(Ω^{n}). More precisely if X is any compact n-cycle, there is a neighborhood U of X such that integration of elements of H^{n}(U, Ω^{n}) on cycles in U gives local coordinates on C^{n}(Z) at X. To obtain the result he first shows that a compact n-dimensional subvariety of a reduced analytic space admits a neighborhood basis of n-complete open sets.

D. Lieberman (Princeton, N.J.)

MR0592784 (82d:32013) 32G10 14C05 32J25 Barlet, Daniel Familles analytiques de cycles et classes fondamentales relatives. (French) Functions of several complex variables, IV (Sem. François Norguet, 1977–1979) (French), pp. 1–24, Lecture Notes in Math., 807, Springer, Berlin, 1980. Given a complex manifold Z and a (flat) family of compact cycles {X_{s}: s ∈ S} of Z, the author constructs a relative cycle class in H_{|X|}^{p}(S × Z, Ω_{S×Z/S}^{n+p}). The construction is direct and geometric, avoiding dualizing complexes, etc. (When S is a point, one obtains a simple construction of the cohomology class of a cycle: utilizing the local representation of a variety as a branched cover of a polydisc, i.e., a family of zero cycles of one polydisc parametrized by another, one reduces to finding the cycle class of the universal Chow zero cycle, which is handled by explicit construction.)

D. Lieberman (Princeton, N.J.)

MR0603157 (81m:32027) 32G11 32B99 41A60 Barlet, Daniel Développement asymptotique des fonctions obtenues par intégration sur les fibres. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 11, A611–A614. Author’s summary: “If f: X → S is a pure-dimensional morphism of a complex reduced analytic space X on the unit disc of C, we show that the function obtained by integration over the fibres of f of a C^{∞}-form with proper support and appropriate type, has an asymptotic expansion when s → 0. We give ‘explicitly’ the first nonconstant term of this expansion in the case where f^{−1}(0) is smooth with multiplicity k ≥ 3, by introducing the symbol of the order-one deformation associated to the family of the fibers of f at s = 0.”

MR0556441 (80j:32014) 32B99 32C25 32G05 Barlet, Daniel Déformations d’ordre un de cycles analytiques. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 12, A617–A620. Author’s summary: “This note contains results on first-order deformations of analytic cycles of a smooth, complex analytic manifold. We show that, in general, the first-order jet of a one-parameter family of cycles does not determine the first-order jet of its graph. But we give a generic situation in which this is nevertheless true.”

MR0522012 (80b:32012) 32C45 Barlet, Daniel Majoration du volume des fibres génériques et forme géométrique du théorème d’aplatissement. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 1, A29–A31. Author’s summary: “We show how the bound of the volume of the generic fibers of a proper surjective morphism between reduced irreducible analytic spaces is involved in geometric flattening of this morphism.”

MR0518045 (80m:32018) 32E10 32F99 Barlet, Daniel Convexité de l’espace des cycles. (French. English summary) Bull. Soc. Math. France 106 (1978), no. 4, 373–397. Let X be a complex manifold of dimension n. X is said to be strictly q-pseudoconvex if X is a relatively compact open subset of another complex manifold and if locally the boundary of X is defined by the vanishing of a real-valued C^{2} function ψ such that dψ ≠ 0 and the restriction of the Levi form of ψ to the holomorphic tangent space of the boundary of X has n − q − 1 positive eigenvalues and q negative eigenvalues. X is said to be strongly q-pseudoconvex if there exists a C^{2} exhaustion function ψ on X such that for some compact subset K of X, the Levi form of ψ has at least n − q positive eigenvalues at every point of X ∖ K. X is said to be q-complete if X is strongly qpseudoconvex and the compact subset of K in the preceding definition can be chosen to be the empty set. (Note: All strictly q-pseudoconvex manifolds are strongly q-pseudoconvex.) Denote by C_{q}(X) the space of all formal finite linear combinations of irreducible q-dimensional compact subvarieties of X with positive integers as coefficients. It was pointed out by A. Andreotti and F. Norguet [Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 31–82; MR0239118 (39 #477)] that, if X is an open subset of the regular points of a subvariety of CP_{N}, then C_{q}(X) admits a complex analytic structure. In the same paper they proved that, if X is strictly q-pseudoconvex, then C_{q}(X) is holomorphically convex; furthermore, if H^{q+1}(X, ℱ) = 0 for any coherent analytic sheaves ℱ over X, then C_{q}(X) is a Stein space. In a later paper, Norguet and Y. T. Siu [Bull. Soc. Math. France 105 (1977), no. 2, 191–223; MR0590090 (58 #28677)], among some other things, extended this theorem with the assumptions that X is an open subset of the regular points of a subvariety of CP_{N} and X is strongly q-pseudoconvex. It should be mentioned that in a previous paper of the author [Fonctions de plusieurs variables complexes, II (Sem. François Norguet, 1974/1975), pp. 1–158, Lecture Notes in Math., Vol. 482, Springer, Berlin, 1975; MR0399503 (53 #3347)] the analytic space C_{q}(X) was constructed for any complex analytic space X. In the present paper, using an L^{2}-estimate for the \overline{∂}-operator and the Norguet-Siu generalization of the Levi problem the author proves the following results for a more general analytic space X. Theorem 1: Suppose X is a finite-dimensional complex analytic space which is q-complete; then C_{q}(X) is Stein. Theorem 2: Suppose X is a finitedimensional complex analytic space which is strongly q-pseudoconvex; then C_{q}(X) is holomorphically convex.

B. Wong (Princeton, N.J.)

MR0521919 (80i:32037) 32C15 Barlet, Daniel Le faisceau ω_{X}^{·} sur un espace analytique X de dimension pure. (French) Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977), pp. 187–204, Lecture Notes in Math., 670, Springer, Berlin, 1978. Let X be a reduced complex space of pure dimension n, Ω_{X}^{·} its sheaf of holomorphic differential forms and j: S → X the inclusion of the singular sets of X. The author constructs a graded sheaf ω_{X}^{·} on X with the following important properties: (a) ω_{X}^{·} is a coherent subsheaf of j_{∗}j^{∗}Ω_{X}^{·} and ω_{X}^{n} is isomorphic to the Grothendieck sheaf of X. (b) Given any local parametrization of X at x_{0} (that is, a proper and finite holomorphic mapping f of a neighborhood of x_{0} onto a polydisc U in C^{n}), the “trace” of ω̃ ∧ ν by f, defined as expected using the ramification set of f in U, is a holomorphic germ on U, for all germs ω̃ ∈ ω_{X,x_{0}}^{·} and ν ∈ Ω_{X,x_{0}}^{·}. (c) More generally, ω_{X}^{·} behaves covariantly under proper, finite, surjective morphisms of complex spaces. (d) ω_{X}^{·} is a canonical direct factor of the complex of d′′-closed currents of X of bidegree (·, 0), a section being given by the multiple-residue operators of Coleff, Lieberman and the reviewer. (e) If X is Cohen-Macaulay and Y is an analytic subset of X of pure codimension q, a fundamental class of Y can be constructed, with values in Ext_{𝒪_{X}}^{q} (𝒪_{Y} , ω_{X}^{q}).

M. Herrera (Buenos Aires)

MR0590081 (58 28671) 32F10 Barlet, Daniel Base de voisinages n-complets pour un sous-ensemble analytique compact de dimension n. Applications. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 17, A751–A753. Author’s summary: “We show that a compact n-dimensional analytic subset of a finite-dimensional analytic complex space has a fundamental system of open neighbourhoods which are n-complete (in the sense of Andreotti-Grauert). We give some applications of this result, for instance we obtain a ‘concrete’ local embedding of the space of compact analytic cycles of a complex analytic manifold (à la ‘Andreotti-Norguet’).”

MR0589935 (58 28649) 32C35 Barlet, Daniel Faisceau ω_{X}^{·}, trace universelle et courants holomorphes sur un espace analytique réduit X de dimension pure. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 11, Aii, A579–A582. Author’s summary: “On a reduced pure-dimensional analytic space X we introduce a (coherent) graded sheaf ω_{X}^{·} which coincides with the sheaf of differential forms Ω_{X}^{·} at the smooth points of X. We obtain a characterization of the sections of ω_{X}^{·} in terms of traces, and from it we deduce that the sheaf ω_{X}^{·} is identified with the quotient of the sheaf of holomorphic currents on X by its torsion subsheaf.”

MR0447633 (56 5943) 32F10 32E10 Barlet, Daniel Convexité de l’espace des cycles. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 15, Ai, A1021–A1023. The following result is announced: Let X be a q-complete complex analytic space of bounded dimension; then the complex analytic space 𝒞_{q}(X) of the compact cycles of X of pure dimension q is a Stein space. This generalizes deep results of A. Andreotti and F. Norguet [Ann. Scuola Norm Sup. Pisa (3) 21 (1967), 31–82; MR0239118 (39 #477)] in which the result was obtained for smooth X open in a projective algebraic variety and those of Norguet and Y. T. Siu (“Holomorphic convexity of spaces of analytic cycles”, Bull. Soc. Math. France, to appear) in which the result was obtained for X as above but only strongly q-pseudoconvex. It must be mentioned that the techniques exploited by the author are different. In particular no use is made of the vanishing theorems of Andreotti and H. Grauert [Bull. Soc. Math. France 90 (1962), 193–259; MR0150342 (27 #343)].

Alessandro Silva (Stony Brook, N. Y.)

MR0399503 (53 3347) 32C25 32G13 Barlet, Daniel Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie. (French) Fonctions de plusieurs variables complexes, II (Sém. François Norguet, 1974–1975), pp. 1–158. Lecture Notes in Math., Vol. 482, Springer, Berlin, 1975. The author develops (in 158 pages) a theory to provide the set of compact analytic n-dimensional cycles on a given finite-dimensional analytic space with the structure of a reduced finite-dimensional analytic space. These cycles are described locally as k-fold coverings of C^{n}, or equivalently, as maps into the k-fold symmetric product of C^{n}. The change of projection creates difficulties, which make it not evident how to define analytic families of cycles. But once this is done, the problem is posed in the usual way as representing the functor associating to a space S the analytic families of cycles parametrized by S. The contents of the chapters: (0) Preliminaries. (I) On pose le problème. (II) Isotropie. (III) Construction of the space of cycles. (IV) Direct images—Chow and cycles. (V) Mapping Douady into cycles. (VI) Intersection theory with parameters. (VII) Integration of cohomology classes.

{For the entire collection see MR0377095 (51 #13269).} W. Barth

MR0382728 (52 3610) 32G10 Barlet, Daniel Familles analytiques de cycles et classes fondamentales relatives. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Ai, A1443–A1446. The author describes the construction of a fundamental class c_{X}^{Z} ∈ H^{p,p}(Z) for an analytic cycle X in a complex manifold Z of pure codimension p in elementary geometric terms, namely by using k-fold symmetric products of C^{p}. He extends this to the relative case, i.e., to analytic families of cycles in Z. For details he refers to another article [Fonctions de plusieurs variables complexes, II (Sém. François Norguet, Janvier 1974-Juin 1975), pp. 1–158, Lecture Notes in Math., Vol. 482, Springer, Berlin, 1975].

W. Barth

MR0442292 (56 678) 32G13 14C05 Barlet, Daniel Cycles dépendant analytiquement d’un paramètre. (French) C. R. Acad. Sci. Paris Sér. A 278 (1974), 1123–1125. Let X be a family of n-dimensional compact cycles of a complex manifold Z, parametrized by a finite-dimensional reduced complex space S. In a previous note [same C. R. Ser. A 278 (1974), A557-A559; MR0361168 (50 #13614)], the author gave sufficient conditions for the analyticity of X at s_{0} ∈ S, to be verified on all members of all adapted coverings of the support |X_{s_{0}}| of X_{s_{0}}. Stronger local conditions are given in this note that assure analyticity of X at s_{0}, provided they are fulfilled on just one adapted covering of |X_{s_{0}}|.

M. Herrera MR0369738 (51 5970) 32F15 14C99 Barlet, Daniel Espace des cycles et d′d′′-cohomologie de P_{n} − P_{k}. (French) Fonctions de plusieurs variables complexes (Sém. François Norguet, 1970-1973; à la mémoire d’Andre Martineau), pp. 98–123. Lecture Notes in Math, Vol. 409, Springer, Berlin, 1974. Let C_{d}^{+}(X) (X = P_{n} − P_{k}, k < n − 1) be the space of d-dimensional compact analytic cycles on X. An integration defines the map r_{0}: H^{d}(X, Ω^{d}) → Γ(C_{d}^{+}(X), 𝒪). Let V^{d,d}^{(X)} be the space of pluriharmonic forms φ on X such that ∂\overline{∂}φ = 0 and φ = ∂a + \overline{∂}b, a and b being C^{∞}-forms on X. Denote by H(C_{d}^{+}(X)) the space of pluriharmonic functions on C_{d}^{+}(X). Then an integration defines the map r_{1}: V^{d,d}(X) → H(C_{d}^{+}(X)). The author proves that r_{0} and r_{1} are isomorphisms. The maps r_{0} and r_{1} were constructed by A. Andreotti and F. Norguet [Ann. Scuola Norm. Sup. Pisa (3) 25 (1971), 59–114; MR0288314 (44 #5512)].

{For the entire collection, see MR0346176 (49 #10902).} B. Cenkl

MR0361168 (50 13614) 32G99 32H99 Barlet, Daniel Cycles dépendant analytiquement d’un paramètre. (French) C. R. Acad. Sci. Paris Sér. A 278 (1974), 557–559. Soit Z une variété C-analytique de dimension n + p. Une écaille E = (f, U, B) dans Z est un isomorphisme analytique d’un ouvert V de Z sur un voisinage ouvert de \overline{U} × \overline{B} dans C^{n+p}, où U et B sont des polydisques ouverts et relativement compacts de C^{n} et C^{p}, respectivement. Soit X = ∑_{i∈I} n_{i}X_{i} un cycle analytique compact de dimension n de Z. On dit que E est adaptée à X si X_{i} ∩f^{−1}(\overline{U} ×δB) = ∅ (i ∈ I). Dans ces conditions, l’auteur associe à X un degré k dans E, et une application analytique f_{E}(X): U → sym^{k}(B), où sym^{k}(B) note le quotient de B^{k} par l’action du groupe des permutations de l’ensemble à k elements, construite en regardant f(X_{i}) ∩ (U × B) comme revêtement ramifié de degré k_{i} de U(i ∈ I), et en définissant k = ∑_{I} k_{i}n_{i}.

Soit X_{s} (s ∈ S) une famille de cycles compacts de dimension n de Z, où S est un espace analytique réduit de dimension finie. Cette famille dépend analytiquement de s au voisinage de s_{0} ∈ S, si pour toute écaille E de Z adaptée à chacun des X_{s} pour chaque s d’un voisinage ouvert W_{E} de s_{0} on a: (i) le degré k de X_{s} dans E ne dépend pas de s ∈ W_{E}; (ii) l’application W_{E} × U → sym^{k}(B), (s, u) → f_{E}(X_{s})(u), est analytique.

L’auteur énonce des properiétés générales de cette notion de dépendance analytique et montre qu’il ne suffit pas de vérifier (i) et (ii) sur une famille d’écailles recouvrant le support de X_{s_{0}}.

M. Herrera