On Lelong Numbers of Generalized Monge-Amp\`ere Products

TL;DR

This paper studies generalized Monge-Ampère products of quasiplurisubharmonic functions, providing estimates for their Lelong numbers under pushforward operations, and applies these results to analyze the positivity properties of vector bundles.

Contribution

It extends the understanding of Lelong number estimates for generalized Monge-Ampère products and applies these to pseudoeffective vector bundles, generalizing recent results on nefness.

Findings

Established bounds for Lelong numbers of pushforward Monge-Ampère products.

Applied estimates to Chern and Segre currents of vector bundles.

Proved a criterion for nefness based on non-nef locus and Chern class conditions.

Abstract

We consider generalized (mixed) Monge-Amp\`ere products of quasiplurisubharmonic functions (with and without analytic singularities) as they were introduced and studied in several articles written by subsets of M. Andersson, E. Wulcan, Z. B{\l}ocki, R. L\"ark\"ang, H. Raufi, J. Ruppenthal, and the author. We continue these studies and present estimates for the Lelong numbers of pushforwards of such products by proper holomorphic submersions. Furthermore, we apply these estimates to Chern and Segre currents of pseudoeffective vector bundles. Among other corollaries, we obtain the following generalization of a recent result by X. Wu. If the non-nef locus of a pseudoeffective vector bundle $E$ on a K\"ahler manifold is contained in a countable union of $k$-codimensional analytic sets, and if the $k$-power of the first Chern class of $E$ is trivial, then $E$ is nef.