Volumes of components of Lelong upper level sets II

TL;DR

This paper establishes optimal upper bounds for the volumes of components of Lelong upper level sets of positive currents on compact Kähler manifolds, linking geometric measure estimates to cohomological data.

Contribution

It introduces a new method to bound the volume of Lelong level set components using cohomology classes of self-products of positive currents.

Findings

Derived optimal volume bounds for Lelong level set components.

Connected geometric measure estimates with cohomological invariants.

Extended previous results to a broader class of currents.

Abstract

Let $X$ be a compact K\"ahler manifold of dimension $n$, and let $T$ be a closed positive $(1,1)$-current in a nef cohomology class on $X$. We establish an optimal upper bound for the volume of components of Lelong upper level sets of $T$ in terms of cohomology classes of non-pluripolar self-products of $T$.