Residue currents associated with weakly holomorphic functions

TL;DR

This paper develops residue currents for weakly holomorphic functions, extending classical residue theory to broader contexts, and demonstrates their fundamental properties and equivalences in complex analysis.

Contribution

It introduces residue currents for weakly holomorphic functions, establishing key properties and their relations to classical residue currents in complex geometry.

Findings

Residue currents satisfy transformation laws similar to the strongly holomorphic case.

The Poincaré-Lelong formula holds for these residue currents.

Equivalence between Coleff-Herrera products and Bochner-Martinelli currents is established for complete intersections.

Abstract

We construct Coleff-Herrera products and Bochner-Martinelli type residue currents associated with a tuple $f$ of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case, as the transformation law, the Poincar\'e-Lelong formula and the equivalence of the Coleff-Herrera product and the Bochner-Martinelli type residue current associated with $f$ when $f$ defines a complete intersection.