Restricted volumes and divisorial Zariski decompositions

TL;DR

This paper explores the connection between Zariski decompositions and restricted volumes of big divisors on complex projective varieties, extending the concept to transcendental classes on Kähler manifolds with an analytic approach.

Contribution

It establishes a link between Zariski decompositions and restricted volumes and introduces an analytic definition of restricted volume for transcendental classes.

Findings

Relation between Zariski decomposition and restricted volume

Analytic description of restricted volume

Extension to transcendental classes on Kähler manifolds

Abstract

We give a relation between the existence of a Zariski decomposition and the behavior of the restricted volume of a big divisor on a smooth (complex) projective variety. Moreover, we give an analytic description of the restricted volume in the line of Boucksom's work. It enables us to define the restricted volume of a transcendental class on a compact K\"ahler manifold in natural way. The relation can be extended to a transcendental class.