Green functions, Segre numbers, and King's formula

2013/04/29 by Andersson, Mats, Wulcan, Elizabeth · 1 citation
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1304.7675

Abstract

Let \mathcal J be a coherent ideal sheaf on a complex manifold X with zero set Z, and let G be a plurisubharmonic function such that G=log|f|+\mathcal O(1) locally at Z, where f is a tuple of holomorphic functions that defines \mathcal J. We give a meaning to the Monge-Ampère products (dd^c G)^k for k=0,1,2,..., and prove that the Lelong numbers of the currents M_k^{\mathcal J}:=\mathbf 1_Z(dd^c G)^k at x coincide with the so-called Segre numbers of \mathcal J at x, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that M_k^{\mathcal J} satisfy a certain generalization of the classical King formula.

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• Segre forms of singular metrics on vector bundles and Lelong numbers

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