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Segre numbers, a generalized King formula, and local intersections

Journal article
Authors Mats Andersson
Håkan Samuelsson
Elizabeth Wulcan
Alain Yger
Published in Journal für die Reine und Angewandte Mathematik
Volume 728
Pages 105–136
ISSN 0075-4102
Publication year 2017
Published at Department of Mathematical Sciences
Pages 105–136
Language en
Links dx.doi.org/10.1515/crelle-2014-0109
Subject categories Mathematical Analysis, Geometry

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– 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.

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