Segre numbers, a generalized King formula, and local intersections

Abstract

Abstract Let 𝒥 {{\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 (d d c d^{c} log | f | 2 |f|^{2} ) k )^{k} , where f is a tuple of generators of 𝒥 {{\mathcal{J}}} , coincide with the so-called Segre numbers of 𝒥 {{\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 𝒥 {{\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.