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A class of non-algebraic threefolds

Matei Toma (1989)

Annales de l'institut Fourier

Let X be a compact complex nonsingular surface without curves, and E a holomorphic vector bundle of rank 2 on X . It turns out that the associated projective bundle P E has no divisors if and only if E is “strongly” irreducible. Using the results concerning irreducible bundles of [Banica-Le Potier, J. Crelle, 378 (1987), 1-31] and [Elencwajg- Forster, Annales Inst. Fourier, 32-4 (1982), 25-51] we give a proof of existence for bundles which are strongly irreducible.

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 converse to the Andreotti-Grauert theorem

Jean-Pierre Demailly (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

The goal of this paper is to show that there are strong relations between certain Monge-Ampère integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of holomorphic line bundles. Especially, we prove that these relations hold without restriction for projective surfaces, and in the special case of the volume, i.e. of asymptotic 0 -cohomology, for all projective manifolds. These results can be seen as a partial converse to the Andreotti-Grauert...

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 ) < ∞ .

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