The growth of dimension of cohomology of semipositive line bundles on Hermitian manifolds

TL;DR

This paper investigates the asymptotic growth of cohomology dimensions for semipositive line bundles on Hermitian manifolds, providing estimates and vanishing theorems applicable to various complex geometric settings.

Contribution

It introduces new asymptotic estimates for cohomology dimensions of semipositive line bundles on Hermitian manifolds under the fundamental estimate, extending to several classes of complex manifolds.

Findings

Asymptotic estimate for harmonic (0,q)-forms with high tensor powers

Dimension bounds for cohomology on q-convex and pseudo-convex manifolds

Vanishing theorems for cohomology on compact manifolds with singular metrics

Abstract

In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic $(0,q)$-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we estimate the dimension of cohomology of semipositive line bundles on $q$-convex manifolds, pseudo-convex domains, weakly $1$-complete manifolds and complete manifolds. We also obtain the estimate of cohomology on compact manifolds with semipositive line bundles endowed with a Hermitian metric with analytic singularities and the related vanishing theorems.