Division Theorems for Exact Sequences

TL;DR

This paper establishes division theorems for exact sequences of holomorphic vector bundles under specific conditions, extending classical results like Skoda's theorem through new geometric and integrability assumptions.

Contribution

It introduces a framework for division theorems in the context of holomorphic vector bundles, enhancing previous results especially for Koszul complexes and singular Hermitian structures.

Findings

Proves division theorems under integrability and geometric conditions

Extends results to Koszul complex cases

Recovers Skoda's division theorem for holomorphic functions

Abstract

Under certain integrability and geometric conditions, we prove division theorems for the exact sequences of holomorphic vector bundles and improve the results in the case of Koszul complex. By introducing a singular Hermitian structure on the trivial bundle, our results recover Skoda's division theorem for holomorphic functions on pseudoconvex domains in complex Euclidean spaces.