Asymptotic cohomology vanishing and a converse to the Andreotti-Grauert theorem on a surface

Mathematics – Complex Variables

Scientific paper

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17 pages

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Scientific paper

Abstract

We study a relation between the curvature positivity and the asymptotic behavior of the higher cohomology for tensor powers of a holomorphic line bundle. A theorem of Andreotti and Grauert asserts that a partial curvature positivity implies asymptotic vanishing of certain higher cohomology groups. We study the converse implication under various situations, for example when a line bundle is semi-ample or big. We prove the converse implication on smooth projective surfaces without any assumptions on a line bundle.

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