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

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.