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A converse to the Andreotti-Grauert theorem. (English. French summary) Zbl 1228.32020
Summary: 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 vanishing theorem.

##### MSC:
 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 14C20 Divisors, linear systems, invertible sheaves 14F99 (Co)homology theory in algebraic geometry
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