The Role of the Gradient Term of the Bochner-Kodaira Formula in Coherent Sheaf Extension

TL;DR

This paper explores the use of the gradient term in the Bochner-Kodaira formula to construct holomorphic sections, leading to new extension results for vector bundles with specific curvature conditions.

Contribution

It introduces a novel application of the gradient term in the Bochner-Kodaira formula to prove extension theorems for holomorphic vector bundles with $L^p$ curvature.

Findings

Constructed holomorphic sections for vector bundles with $L^p$ curvature

Proved Thullen-type extension across codimension 1

Highlighted a new method for using the gradient term in complex geometry

Abstract

In applying the Bochner-Kodaira formula with boundary term to solve the $\bar\partial$ equation with $L^2$ estimates, the gradient term is usually not used. Two potentially important applications of the use of the gradient term are the strong rigidity for holomorphic vector bundles and the very ampleness part of the Fujita conjecture. In this note we use the gradient term to construct holomorphic sections to prove the Thullen-type extension across codimension $1$ for holomorphic vector bundles with Hermitian metric whose curvature is $L^p$ for some $p>1$. This construction of sections points out a typical way of how the gradient term can be used.