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

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9 Citations (Scopus)

Abstract

In this paper, we study relations between positivity of the curvature and the asymptotic behavior of the higher cohomology group for tensor powers of a holomorphic line bundle. The Andreotti-Grauert vanishing theorem asserts that partial positivity of the curvature implies asymptotic vanishing of certain higher cohomology groups. We investigate the converse implication of this theorem under various situations. For example, we consider the case where a line bundle is semi-ample or big. Moreover, we show the converse implication holds on a projective surface without any assumptions on a line bundle.

Original languageEnglish
Pages (from-to)2199-2221
Number of pages23
JournalAnnales de l'Institut Fourier
Volume63
Issue number6
DOIs
Publication statusPublished - 2013 Jan 1
Externally publishedYes

Keywords

  • Asymptotic cohomology groups
  • Chern curvatures
  • Hermitian metrics
  • Partial cohomology vanishing
  • Q-positivity

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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