Various approaches to products of residue currents

TL;DR

This paper explores different methods for constructing products of residue currents, analyzing how regularizations of individual currents can produce regularizations of their Coleff-Herrera products on complex spaces.

Contribution

It provides a comprehensive study of approaches to product residue currents and their regularizations, extending results to arbitrary pure-dimensional complex spaces.

Findings

Regularizations of individual residue currents can produce regularizations of their Coleff-Herrera products.

Results are valid globally on arbitrary pure-dimensional complex spaces.

Various approaches to residue current products are systematically compared.

Abstract

We describe various approaches to Coleff-Herrera products of residue currents $R^j$ (of Cauchy-Fantappi\`e-Leray type) associated to holomorphic mappings $f_j$. More precisely, we study to which extent (exterior) products of natural regularizations of the individual currents $R^j$ yield regularizations of the corresponding Coleff-Herrera products. Our results hold globally on an arbitrary pure-dimensional complex space.