# Segre numbers, a generalized King formula, and local intersections Journal article, 2015

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– Ampère 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, Achilles–Manaresi, 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.

## Author

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

#### Håkan Samuelsson Kalm

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

#### Elizabeth Wulcan

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

#### Alain Yger

University of Bordeaux

#### Journal für die Reine und Angewandte Mathematik

0075-4102 (ISSN)

Vol. 728 728 105-136

#### Subject Categories

Mathematics

Geometry

Mathematical Analysis

#### DOI

10.1515/crelle-2014-0109