Decomposition of residue currents

TL;DR

This paper introduces a method to decompose residue currents associated with submodules in complex analysis, linking the decomposition to primary decomposition of modules and developing a class of currents with versatile restriction properties.

Contribution

It provides a novel decomposition technique for residue currents aligned with primary decomposition, and introduces a new class of currents with enhanced restriction capabilities.

Findings

Decomposition of residue currents corresponds to primary decomposition.

Introduction of a new class of currents including residue and principal value currents.

Currents admit restrictions to analytic varieties and constructible sets.

Abstract

Given a submodule $J\subset \mathcal O_0^{\oplus r}$ and a free resolution of $J$ one can define a certain vector valued residue current whose annihilator is $J$. We make a decomposition of the current with respect to Ass$(J)$ that correspond to a primary decomposition of $J$. As a tool we introduce a class of currents that includes usual residue and principal value currents; in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets.