Divergent integrals, residues of Dolbeault forms, and asymptotic Riemann mappings

TL;DR

This paper investigates the asymptotic behavior of divergent integrals involving meromorphic forms on complex manifolds, introducing residues of Dolbeault forms and linking them to Riemann mapping asymptotics.

Contribution

It introduces residues of Dolbeault forms and connects their properties to the asymptotic analysis of divergent integrals on complex manifolds.

Findings

Derived formulas for the asymptotic behavior of divergent integrals

Introduced the concept of residues of Dolbeault forms

Linked integral regularization to Riemann mapping asymptotics

Abstract

We describe the asymptotic behaviour and the dependence on the regularization of logarithmically divergent integrals of products of meromorphic and antimeromorphic forms on complex manifolds. Our formula is expressed in terms of residues of Dolbeault forms, a notion introduced in this paper. The proof is based on a result on the asymptotic behaviour of Riemann mappings of small domains.