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A cohomological Steinness criterion for holomorphically spreadable complex spaces

Viorel Vâjâitu (2010)

Czechoslovak Mathematical Journal

Let X be a complex space of dimension n , not necessarily reduced, whose cohomology groups H 1 ( X , 𝒪 ) , ... , H n - 1 ( X , 𝒪 ) are of finite dimension (as complex vector spaces). We show that X is Stein (resp., 1 -convex) if, and only if, X is holomorphically spreadable (resp., X is holomorphically spreadable at infinity). This, on the one hand, generalizes a known characterization of Stein spaces due to Siu, Laufer, and Simha and, on the other hand, it provides a new criterion for 1 -convexity.

A finiteness theorem for holomorphic Banach bundles

Jürgen Leiterer (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let E be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form id + K where K is compact. Assume that the characteristic fiber of E has the compact approximation property. Let n be the complex dimension of X and 0 q n . Then: If V X is a holomorphic vector bundle (of finite rank) with H q ( X , V ) = 0 , then dim H q ( X , V E ) < . In particular, if dim H q ( X , 𝒪 ) = 0 , then dim H q ( X , E ) < .

Analytic cohomology of complete intersections in a Banach space

Imre Patyi (2004)

Annales de l’institut Fourier

Let X be a Banach space with a countable unconditional basis (e.g., X = 2 ), Ω X an open set and f 1 , ... , f k complex-valued holomorphic functions on Ω , such that the Fréchet differentials d f 1 ( x ) , ... , d f k ( x ) are linearly independant over at each x Ω . We suppose that M = { x Ω : f 1 ( x ) = ... = f k ( x ) = 0 } is a complete intersection and we consider a holomorphic Banach vector bundle E M . If I (resp. 𝒪 E ) denote the ideal of germs of holomorphic functions on Ω that vanish on M (resp. the sheaf of germs of holomorphic sections of E ), then the sheaf cohomology groups H q ( Ω , I ) , H q ( M , 𝒪 E ) vanish...

Corrigendum to: Holomorphic Morse inequalities on manifolds with boundary

Robert Berman (2008)

Annales de l’institut Fourier

A statement in the paper “Holomorphic Morse inequalities on manifolds with boundary” saying that the holomorphic Morse inequalities for an hermitian line bundle L over X are sharp as long as L extends as semi-positive bundle over a Stein-filling is corrected, by adding certain assumptions. A more general situation is also treated.

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