2011/04/28 by Shin‐ichi Matsumura, Matsumura, Shin-ichi · 1 citation
Mathematics · #14C17 #14F17 #32L15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1104.5313
openalex publication_date 2011/04/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.