Segre numbers, a generalized King formula, and local intersections

TL;DR

This paper establishes a deep connection between Segre numbers, generalized Monge-Ampère products, and a generalized King formula, providing new tools for intersection theory on complex analytic spaces.

Contribution

It introduces a new calculus for positive currents and generalizes classical formulas to include fixed and moving components of Vogel cycles.

Findings

Lelong numbers of certain restricted currents equal Segre numbers

Generalized King formula accounts for fixed and moving Vogel cycle components

New calculus for products of Bochner-Martinelli type currents

Abstract

Let $\mathcal J$ be an ideal sheaf on a reduced analytic space $X$ with zero set $Z$. We show that the Lelong numbers of the restrictions to $Z$ of certain generalized Monge-Amp\`ere products $(dd^c\log|f|^2)^k$, where $f$ is a tuple of generators of $\mathcal J$, coincide with the so-called Segre numbers of $\mathcal J$, introduced independently by Tworzewski and Gaffney-Gassler. More generally we show that these currents satisfy a generalization of the classical King formula that takes into account fixed and moving components of Vogel cycles associated with $\mathcal J$. A basic tool is a new calculus for products of positive currents of Bochner-Martinelli type. We also discuss connections to intersection theory.