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MR0511894 (80c:32025) 32G05 Donin, I. F. Deformations of noncompact complex spaces. (Russian) Uspekhi Mat. Nauk 33 (1978), no. 5(203), 183–184. The aim of the paper is the following theorem. Let X be a strongly 1-pseudoconvex complex space. Then X possesses a locally trivial finite-dimensional effective deformation which is versal among all locally trivial deformations of any holomorphically convex compact subset of X. If the sheaf Ex^{1}(X) has a compact support then there exists a finite-dimensional effective deformation of X, which is versal among all deformations of any holomorphically convex compact part of X. The proof (which is not given) uses deformations with Banachanalytic bases and theorems on the continuation of an imbedding of a fibre which is Stein to a neighborhood of the fibre.

K. Wolffhardt (Munich)

MR0393582 (52 14391) 32G13 Donin, I. F. Construction of versal deformations for projective algebraic varieties. (Russian) Mat. Sb. (N.S.) 95(137) (1974), 19–34, 159. On démontre l’existence d’une famille verselle de déformations pour les variétés algébriques projectives.

{English translation: Math. USSR-Sb. 24 (1974), no. 1, 17–30 (1976).}

J.-M. Kantor

MR0393581 (52 14390) 32G13 14D15 Donin, I. F. Construction of a versal family of deformations for holomorphic principal bundles over a compact complex space. (Russian) Mat. Sb. (N.S.) 94(136) (1974), 430–443, 496. Il est démontré un théorème d’existence de déformation verselle pour une fibration holomorphe sur un espace compact (la fibre étant un groupe de Lie).

{English translation: Math. USSR-Sb. 23 (1974), no. 3, 405–416 (1975).}

J.-M. Kantor MR0590099 (58 28681) 32G13 Donin, I. F. Versal families of deformations of holomorphic bundles. (Russian) Uspehi Mat. Nauk 28 (1973), no. 5(173), 239–240. The author states results which show the existence of a complete family of versal deformations of holomorphic principal bundles over a compact complex space with Lie structure group. The problem in the bundle case vis-à-vis the absolute case is that one has both a problem of privileged coverings and a nonabelian cohomology problem. Insufficient detail is given to determine the author’s priorities. The result has been obtained by H. Grauert [Invent. Math. 25 (1974), 107–142; MR0346194 (49 #10920)], who emphasized the nonabelian cohomology, and by O. Forster and K. Knorr [Math. Ann. 209 (1974), 291–346; MR0374495 (51 #10695); Invent. Math. 16 (1972), 113–160; MR0325998 (48 #4344); Konstruktion verseller Familien kompakter komplexer Räume, Lecture Notes in Math., Vol. 705, Springer, Berlin, 1979] who make a beautiful extension of elementary methods to the relative case. Their method is less technical in the absolute case, and the essence can be gleaned from Forster’s Williamstown lectures [Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975), pp. 199–217. Amer. Math. Soc., Providence, R.I., 1977; MR0470250 (57 #10008)]. For details see the article by the author [Mat. Sb. (N.S.) 94(136) (1974), 430–443; MR0393581 (52 #14390)].

J. S. Joel (Ann Arbor, Mich.)

MR0333249 (48 11574) 32G10 Donin, I. F. Complete families of deformations of germs of complex spaces. (Russian) Mat. Sb. (N.S.) 89(131) (1972), 390–399, 533. The author proves that every analytic space X with dim_{C} Ex^{1}(X) < ∞ has a finite-dimensional complete family of deformations. (If X is reduced, then Ex^{1}(X) = Ext_{𝒪_{X,x}}^{1} (Ω_{X,x}^{1}, 𝒪_{X,x}).)

{This article has appeared in English translation [Math. USSR-Sb. 18 (1972), 397–406].}

G. K. Kiremidjian MR0299829 (45 8877) 32C35 Donin, I. F. Cohomology with estimates for coherent analytic sheaves over complex spaces. (Russian) Mat. Sb. (N.S.) 86(128) (1971), 339–366. The author studies a coherent sheaf ℱ on a complex analytic space X corresponding to a covering V = {V_{i}} of X with pseudoconvex neighbourhoods. If a pseudo-norm is defined on the cochain space C^{∗}(V, ℱ) such that for every p the elements of C^{p}(V, ℱ) with finite norm form a Banach space C_{B}^{p}(V, ℱ), one may ask whether a coboundary c ∈ C_{B}^{p}(V, ℱ) is also the coboundary of an element c′ ∈ C_{B}^{p−1}(V, ℱ) and whether the norm of c′ can be estimated in terms of the norm of c. Usually it will not be so, but the author proves that it is very often (and always if X is compact) possible to introduce a continuous family V_{ε} = {V_{iε}} of coverings of X such that V_{ε} grows finer in a continuous way when ε decreases, and for ε_{1} < ε_{2} it will then be possible to give an estimate of c′ in terms of c. With supremum norms the estimate will have the form ∥c′∥ ≤ K(ε_{2} − ε_{1})^{−α}∥c∥, where K and α are positive constants depending only on ℱ and the family V_{ε}. The principal tool is a version of Weierstrass’ preparation theorem with improved estimates.

H. Tornehave

MR0277758 (43 3491) 32.47 46.00 Donin, I. F. Banach analytic spaces, and the space of modules of holomorphic bundles. (Russian) Dokl. Akad. Nauk SSSR 195 1970 1010–1013 For an analytic fibre bundle P over a compact complex analytic space X whose structure group is some given complex Lie group, there exists (locally) a complete family of deformations, the module space of which has embedding dimension dim H^{1}(X, Ad P). The proof of this theorem involves interesting considerations on subspaces of a Banach analytic space that have a finite-dimensional tangent space at a given point. The following is a corollary of a more general theorem. A “complete” germ of Banach analytic spaces with a finite-dimensional tangent space contains a largest finite-dimensional analytic subgerm. (A complete germ is one that is isomorphic to a germ of a fiber of a holomorphic map of Banach manifolds, where the associated homomorphism of tangent spaces is strict.)

{This article has appeared in English translation [Soviet Math.

Dokl. 11 (1970), 1591–1594].} K. Wolffhardt

MR0259959 (41 4588) 57.70 32.00 Donin, I. F. Conditions for the triviality of deformations of complex structures. (Russian) Mat. Sb. (N.S.) 81 (123) 1970 610–621 Let X and Y be (not necessarily reduced) complex spaces. A holomorphic map f: X → Y will be called a morphism. A morphism f is said to be flat if and only if 𝒪_{x} is a flat module over 𝒪′_{f(x)} for every x ∈ X, where 𝒪 [𝒪′] denotes the structure sheaf of X [Y]. For a complex space S, an S-space is by definition a complex space X with a morphism π: X → S. If X = (X, π_{1}) and Y = (Y, π_{2}) are S-spaces, a morphism f: Y → Y is called an S-morphism if and only if π_{1} = π_{2} ∘ f. A morphism f: X → Y is called an imbedding if and only if f is an isomorphism of X onto a complex subspace of Y. If f: X → Y is an Smorphism of X = (X, π_{1}) into Y = (Y, π_{2}), then for a point x ∈ X, f induces a morphism f_{s}: X_{s} → Y_{s} of the complex space X_{s} = π_{1}^{−1}(s) into Y_{s} = π_{2}^{−1}(s), where s = π_{1}(x). Put y = f(x). The author first proves the following proposition: Suppose that π_{1} is flat and that f_{s} is an isomorphism of X_{s} onto Y_{s}; then there exist neighborhoods U of x and V of y such that f|_{U}: U → V is an S′-isomorphism with S′ = π_{1}(U) = π_{2}(V). Next, the author proves that if X is a compact complex space and f: X → X is an imbedding, then f is an isomorphism. After preparing several lemmas, the author proves the following main theorem: Let f: X → S be a proper flat morphism and suppose that X_{s} is isomorphic to X_{0} for every s ∈ S, where X_{s} = f^{−1}(s). Then X is trivial over S, i.e., X is a fibre bundle with fibre X_{0}, projection f and structure group Aut(X_{0}). This theorem generalizes a result of W. Fischer and H. Grauert [Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. II (1965), 89–94; MR0184258 (32 #1731)]. Moreover, the author proves the following: Let f: X → S be a proper morphism and suppose that S is a reduced complex space and that X_{s} is isomorphic to X_{0} for every s ∈ S. Then X is trivial over S. Finally, the author obtains the following theorem: Let f: X → S be a locally trivial deformation of a compact complex space X_{0} with 0 ∈ S, and suppose that H^{1}(X_{0}, Ω_{0}) = 0, where Ω_{0} denotes the sheaf of germs of holomorphic vector fields on X_{0}. Then there exists a neighborhood U of 0 in S such that X|_{U} is isomorphic to X_{0} × U.

A. Morimoto MR0237821 (38 6102) 32.40 Donin, I. F. Conditions for the triviality of deformations of holomorphic fibre bundles over a compact complex space. (Russian) Mat. Sb. (N.S.) 77 (119) 1968 602–623 Sei S ein reduzierter komplexer Raum, 0 ∈ S, X ein reduzierter kompakter komplexer Raum und G eine komplexe Lie-Gruppe. 𝒫 sei eine holomorphe Familie von Prinzipalbündeln über X mit der Strukturgruppe G, d.h. ein holomorphes Prinzipalbündel über X × S mit der Strukturgruppe G. 𝒫 heißt trivial, wenn 𝒫 zu dem über X × S vermöge der kanonischen Projektion X × S → X von einem Bündel über X induzierten Bündel äquivalent ist. Für s ∈ S bezeichne 𝒫_{s} das von 𝒫 vermöge X → X × S, x ↦ (x, s) über X induzierte Bündel. Das Hauptergebnis der vorliegenden Arbeit ist folgendes zu einem Ergebnis von W. Fischer und H. Grauert [Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1965, 89–94; MR0184258 (32 #1731)] analoge Theorem I: Genau dann ist die Familie 𝒫 über einer Umgebung von 0 trivial, wenn die Bündel 𝒫_{s} für alle s einer Umgebung von 0 äquivalent sind. Daraus folgt Theorem II: Ist H^{1}(X, Ad 𝒫_{0}) = 0, so ist 𝒫 über einer Umgebung von 0 trivial. (Ad 𝒫_{0} bezeichnet das zu 𝒫_{0} adjungierte Vektorraumbündel.) Der Beweis erfordert längere Vorbreitungen. In § 1 wird der Hilbertraum der auf einer relativ kompakten analytischen Menge A quadratintegrierbaren holomorphen Funktionen studiert. Es wird gezeigt, daß er gegenüber biholomorphen Abbildungen von \overline{A} invariant ist und daß seine natürliche Einbettung in den Fréchet-Raum der auf A holomorphen Funktionen stetig ist. In § 2 läßt Verfasser Banachsche Lie-Gruppen auf Banachsche Mannigfaltigkeiten wirken. In § 3 wird Räumen von holomorphen Funktionen mit Werten in einer endlich-dimensionalen komplexen Lie-Gruppe die Struktur einer Banachschen Lie-Gruppe aufgeprägt. § 4 untersucht die Parameterabhängigkeit von Eigenschaften von Komplexen von Banachschen Vektorräumen und bringt folgenden für den Beweis von Theorem I grundlegenden Satz. Sei E eine Familie von holomorphen Vektorraumbündeln über X. Dann hängt α(t) = dim_{C} H^{0}(X, E_{t}) von t ∈ S Zariski-oberhalbstetig ab; ist α bei 0 konstant, so gibt es α(0) Schnitte in E über X × S, die für jedes t einer Umgebung von 0 eine Basis von H^{0}(X, E_{t}) induzieren.

K. Wolffhardt MR0208624 (34 8433) 32.47 57.70 Donin, I. F. Deformations of holomorphic vector-fiberings over the Riemann sphere. (Russian) Mat. Sb. (N.S.) 71 (113) 1966 495–502 Let S be the Riemann sphere, U_{0} = S − {∞}, U_{∞} = S − {0} and U_{0,∞} = U_{0} ∩ U_{∞}. A holomorphic vector-bundle on S is determined by a holomorphic map A: U_{0,∞} → GL(n, C). It is known that every such bundle is equivalent to one for which A(z) = diag(z^{k_{1}}, ⋯, z^{k_{n}}), k_{1} ≤ k_{2} ≤ ⋯ ≤ k_{n}. A deformation A_{t}(z) is a holomorphic map K × U_{0,∞} → GL(n, C), where K is a neighborhood of the origin in C^{m}. The main result states that the A_{t}, for t in the complement of some analytic set, can be brought simultaneously to diagonal form. More precisely: For some smaller K_{1} ⊂ K and analytic set T ⊂ K_{1} (of strictly smaller dimension), there are holomorphic maps B: (K_{1} −T)×U_{0} → GL(n, C) and C: (K_{1} − T) × U_{∞} → GL(n, C) such that

B_{t}(z)A_{t}(z)C_{t}(z) = diag(z^{p_{1}}, ⋯, z^{p_{n}}).

Moreover, T is empty if and only if p_{i} = k_{i} for all i.

The paper contains the proofs, whose exposition is careful and clear. The author mentions that the theorem was stated by H. Röhrl [Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1960/61), 435–477; MR0131881 (24 #A1728)], but a gap was pointed out by P. L. Poljakov, whose counter-example to a statement in Röhrl’s proof is given at the end of the paper.

M. A. Kervaire