Volume of components of Lelong upper-level sets

TL;DR

This paper establishes an upper bound on the volume of certain analytic sets related to positive currents and applies this to bound the singular locus volume of analytic sets on compact Kähler manifolds.

Contribution

It provides a new uniform upper bound on the volume of singular loci of analytic sets based on their volume, involving Lelong numbers and positive currents.

Findings

Upper bound for volume of maximal analytic sets with positive Lelong number

Uniform upper bound on singular locus volume in terms of total volume

Application to analytic sets on compact Kähler manifolds

Abstract

We prove an upper bound for the volume of maximal analytic sets on which the generic Lelong number of a closed positive current is positive. As a particular case, we give a uniform upper bound on the volume of the singular locus of an analytic set in terms of its volume on a compact Kahler manifold.