Restricted volumes on K\"ahler manifolds

TL;DR

This paper investigates the properties of numerical restricted volumes of (1,1) classes on compact Kähler manifolds, proposing a conjecture about their behavior on non-Kähler loci and proving it under certain conditions.

Contribution

It introduces a conjecture relating non-Kähler loci to restricted volumes and proves it for classes with a Zariski decomposition, expanding understanding of Kähler geometry.

Findings

Proves the conjecture for classes with Zariski decomposition.

Establishes a link between non-Kähler loci and vanishing restricted volumes.

Provides applications of the main results in Kähler geometry.

Abstract

We study numerical restricted volumes of (1,1) classes on compact Kahler manifolds, as introduced by Boucksom. Inspired by work of Ein-Lazarsfeld-Mustata-Nakamaye-Popa on restricted volumes of line bundles on projective manifolds, we pose a natural conjecture to the effect that irreducible components of the non-Kahler locus of a big class should have vanishing numerical restricted volume. We prove this conjecture when the class has a Zariski decomposition, and give several applications.