A Calculus for Finite Parts and Residues of some Divergent Complex Geometric Integrals

TL;DR

This paper develops a calculus for computing finite parts and residues of divergent complex geometric integrals on singular spaces, using current extensions and explicit decompositions.

Contribution

It introduces a new current calculus and formulas to compute finite parts of divergent integrals on complex spaces with singularities, extending previous methods.

Findings

Decomposition formula for finite parts of integrals into explicit currents

Method to reduce general integrals to a special class for computation

Framework applicable to integrals with singularities along subvarieties

Abstract

We consider divergent integrals $\int_X \omega$ of certain forms $\omega$ on a reduced pure-dimensional complex space $X$. The forms $\omega$ are singular along a subvariety defined by the zero set of a holomorphic section $s$ of some holomorphic vector bundle $E$. Equipping $E$ with a smooth Hermitian metric allows us to define a finite part $\mathrm{fp}\,\int_X \omega$ of the divergent integral as the action of a certain current extension of $\omega$. We introduce a current calculus to compute finite parts for a special class of $\omega$. Our main result is a formula that decomposes the finite part of such an $\omega$ into sums of products of explicit currents. Lastly, we show that, in principle, it is possible to reduce the computation of $\mathrm{fp}\,\int_X \omega$ for a general $\omega$ to this class.