On Monge-Amp\`ere volumes of direct images

TL;DR

This paper investigates the asymptotic behavior of Monge-Ampère volumes of direct images from high tensor powers of ample line bundles, classifies bundles reaching topological bounds, and characterizes vector bundles with projectively flat Hermitian structures.

Contribution

It provides a detailed analysis of Monge-Ampère volume asymptotics and characterizes bundles with special geometric structures, extending Demailly's topological bounds.

Findings

Classification of bundles saturating Demailly's topological bound

Characterization of vector bundles with projectively flat Hermitian structures

Asymptotic formula for Monge-Ampère volumes of direct images

Abstract

This paper is devoted to the study of the asymptotics of Monge-Amp\`ere volumes of direct images associated with high tensor powers of an ample line bundle. We study the leading term of this asymptotics and provide a classification of bundles saturating the topological bound of Demailly. In the special case of high symmetric powers of ample vector bundles, this provides a characterization of vector bundles admitting projectively flat Hermitian structures.