Cohomology dimension growth for Nakano $q$-semipositive line bundles

TL;DR

This paper investigates the asymptotic growth of cohomology dimensions for Nakano q-semipositive line bundles on complex manifolds, providing optimal estimates and extending results to non-compact and almost complex manifolds.

Contribution

It offers new asymptotic estimates for cohomology dimensions of Nakano q-semipositive line bundles, including non-compact and almost complex manifolds, with optimal order of growth.

Findings

Derived asymptotic estimates for cohomology dimensions

Extended estimates to non-compact complex manifolds

Provided bounds for the modified Dirac operator on almost complex manifolds

Abstract

We study the cohomology with high tensor powers of Nakano $q$-semipositive line bundles on complex manifolds. We obtain the asymptotic estimates for the dimension of cohomology with high tensor powers of semipositive line bundles over q-convex manifolds and various possibly non-compact complex manifolds, in which the order of estimates are optimal. Besides, estimates for the modified Dirac operator on Nakano $q$-positive line bundle on almost complex manifolds are given.