Definitions of the volume of a big cohomology class

TL;DR

This paper demonstrates the consistency between two definitions of the volume of a big cohomology class and extends the equality to the numerical restricted volume, clarifying foundational aspects of complex geometry.

Contribution

It establishes the equivalence of two major definitions of volume for big cohomology classes and extends this to the numerical restricted volume, enhancing theoretical understanding.

Findings

Two definitions of volume are shown to be consistent.

The equality extends to the numerical restricted volume.

Clarifies foundational aspects of complex geometry.

Abstract

We elaborate on how two definitions of the volume of a big cohomology class are consistent. The first definition involves taking the absolutely continuous part of a closed positive current, and the second involves the non-pluripolar product. We also describe how a similar equality holds for the numerical restricted volume introduced by Collins and Tosatti.