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# Segre numbers, a generalized King formula, and local intersections

Journal article
Authors Mats Andersson Håkan Samuelsson Elizabeth Wulcan Alain Yger Journal für die Reine und Angewandte Mathematik 728 105–136 0075-4102 2017 Department of Mathematical Sciences 105–136 en dx.doi.org/10.1515/crelle-2014-0109 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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