On Finite Parts of Divergent Complex Geometric Integrals and Their Dependence on a Choice of Hermitian Metric

TL;DR

This paper investigates how the finite parts of divergent complex geometric integrals depend explicitly on the choice of Hermitian metric on a vector bundle over a complex space, providing a formula for this dependence.

Contribution

It introduces an explicit formula describing how the finite part of divergent integrals varies with different Hermitian metrics on the bundle.

Findings

Derived an explicit formula for metric dependence of finite parts.

Established a method to regularize divergent integrals in complex geometry.

Clarified the role of Hermitian metrics in integral regularization.

Abstract

Let $X$ be a reduced complex space of pure dimension. We consider divergent integrals of certain forms on $X$ that are singular along a subvariety defined by the zero set of a holomorphic section of some holomorphic vector bundle $E \rightarrow X$. Given a choice of Hermitian metric on $E$ we define a finite part of the divergent integral. Our main result is an explicit formula for the dependence on the choice of metric of the finite part.