Derivative of volumes of big cohomology classes

TL;DR

This paper establishes a precise relationship between the derivative of volume functions of big cohomology classes and their restricted volumes, revealing geometric properties of non-Kaehler loci on compact Kaehler manifolds.

Contribution

It proves that the derivative of the volume function along a divisor equals the numerical restricted volume, linking volume derivatives to geometric loci.

Findings

Derivative of volume equals restricted volume on divisors

Divisorial components of non-Kaehler locus are in the null locus

Provides new insights into the structure of big classes on Kaehler manifolds

Abstract

We prove that the partial derivative of the volume function of big classes along any real divisor in a compact Kaehler manifold is equal to the numerical restricted volume of that class to the divisor. A consequence of our main result is that the divisorial components of the non-Kaehler locus of a big class lie in fact in the null locus of that class.