Théorèmes de finitude pour la cohomologie des espaces complexes

Andreotti, Aldo; Grauert, Hans

doi:10.24033/bsmf.1581

Cited by 538 publications

(401 citation statements)

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“…But to obtain the vanishing theorem, we need only the generalized Lefschetz theorem with coefficients in C. We remark here that the generalized Lefschetz theorem with coefficients in C is proved by a standard technique using cohomological dimension and de Rham cohomology. The last assertion of the lemma follows from Andreotti and Grauert [1].…”

Section: Theorem (411) Let a Be An Abelian Variety Of Dimension N Dmentioning

confidence: 70%

Some results in the theory of vector bundles

Umemura

1973

Nagoya Mathematical Journal

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We have several definitions of the positivity of a vector bundle, differentiate definitions, an algebro-geometric definition, a topological definition etc. In § 1 we review the definitions and the relations between them. For a line bundle all the definitions are equivalent and every one agrees that they are reasonable. For a vector bundle, however, the definitions are not necessarily equivalent. One of the main results of this paper is the equivalence of the definitions over a complete non-singular curve. The proof is given in §2. We proved this over an elliptic curve in Umemura [18]. In this case the proof was based on Atiyah’s classification. To prove the equivalence over a curve of genus ≥ 2, the fundamental lemma is; A stable bundle of positive degree is positive in the sense of Nakano. The tool used to prove this lemma is the theory of stable bundles due to Narasimhan and Seshadri [11] —they establish a correspondence between stable bundles and certain types of irreducible unitary representations of a Fuchsian group.

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Section: Theorem (411) Let a Be An Abelian Variety Of Dimension N Dmentioning

confidence: 70%

Some results in the theory of vector bundles

Umemura

1973

Nagoya Mathematical Journal

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“…Then ∂Ψ/∂z − w and ∂Ψ/∂w are holomorphic on X. We have an entire holomorphic function G(w) := ∂Ψ/∂z − w. We put x := Re z, y := Im z, u := Re w and v := Im w. We can expand Ψ to Fourier series: If X is strongly 1-convex in the sense of Andreotti and Grauert [2], then dim H 1 (X, O) < ∞. Miyajima [7] considers another type of the ∂∂-equation on strongly pseudoconvex Kähler manifolds.…”

Section: §4 Examples and Related Problemsmentioning

confidence: 99%

“…Now suppose that C n /Γ is of cohomologically finite type. Then by Theorem 3.2, ψ (r,p−1) and ψ (r−1,p) are ∂-cohomologue to some constant forms, that is, there exist a (r − 1, p − 1)-form Ψ (1) and a (p − 1, r − 1)-form Ψ (2) such that…”

Section: §1 Introductionmentioning

confidence: 99%

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-problem on weakly 1-complete Kähler manifolds

Kazama

Takayama

1999

Nagoya Mathematical Journal

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“…Andreotti-Grauert [1]). If moreover X is a g-complete manifold of dimension n and EF is locally free, then one has Ho~r(X, o) x ®EF)=Q by Serre's duality, where co x denotes the canonical sheaf of X and Hl~r denotes the cohomology with compact supports.…”

Section: A For Bymentioning

confidence: 99%

A Vanishing Theorem for Proper Direct Images

Ohsawa

1987

Publ. Res. Inst. Math. Sci.

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Let X be a complex analytic space and 3 a coherent analytic sheaf over X. Jf Xis ^-complete, then H r (X, £F), the r-th cohomology of X with coefficients in 3, vanishes for r > q (cf. Andreotti-Grauert [1]). If moreover X is a g-complete manifold of dimension n and EF is locally free, then one has Ho~r(X, o) x ®EF)=Q by Serre's duality, where co x denotes the canonical sheaf of X and Hl~r denotes the cohomology with compact supports.In [2], A. Andreotti and E. Vesentini established an L 2 4heory on noncompact complex manifolds and showed that the vanishing of Hl~r(X, o) x ®3) is a direct consequence of certain a priori estimate as well as the vanishing of H r (X, 3}. Their approach is of interest since the cohomology vanishing is reduced to the solvability of a ^-equation with uniform estimates related to weighted L 2 -norms of Carleman type.The purpose of the present article is to give a relative version of their theory in the following situation.Let /: X-*S be a morphism between complex analytic spaces and £? a coherent analytic sheaf over X. The q-th direct image sheaf R q f*3! is a sheaf over S defined by (R%£F) x : = lim H q (f-\U) y 3*), where U runs through the u neighbourhoods of x. The #-th proper diiect image sheaf R^EF is defined by (R q f&) x :=limH q g(f-l (U\ 3"), where S denotes the family of supports consisting of the subsets of X on which/is proper.A recent result of K. Takegoshi shows that R q f*a) x =Q for q>Q in the above situation, if X is a complex manifold which is bimeromorphically equivalent to a Stein space.

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Théorèmes de finitude pour la cohomologie des espaces complexes

Cited by 538 publications

References 2 publications

Some results in the theory of vector bundles

Some results in the theory of vector bundles

-problem on weakly 1-complete Kähler manifolds

A Vanishing Theorem for Proper Direct Images

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