On weighted Bochner-Martinelli residue currents

TL;DR

This paper investigates the structure of weighted Bochner-Martinelli residue currents associated with holomorphic germs, providing explicit descriptions in the monomial case and linking the currents to Rees valuations and regular sequences.

Contribution

It offers a detailed description of R^p(f) and its annihilator ideal using Rees valuations, and characterizes when R^p(f) is independent of p for monomial sequences.

Findings

R^p(f) described via Rees valuations of the ideal

Explicit form of R^p(f) for monomial sequences

R^p(f) is p-independent iff f is a regular sequence

Abstract

We study the weighted Bochner-Martinelli residue current R^p(f) associated with a sequence f=(f_1,...,f_m) of holomorphic germs at the origin in C^n, whose common zero set equals the origin, and p=(p_1,..., p_m)\in N^n. Our main results are a description of R^p(f) and its annihilator ideal in terms of the Rees valuations of the ideal generated by (f_1^{p_1},..., f_m^{p_m}) and an explicit description of R^p(f) when f is monomial. For a monomial sequence f we show that R^p(f) is independent of p if and only if f is a regular sequence.