Regularity of the volume function

TL;DR

This paper proves the optimal regularity of the volume function on the big cone of a projective manifold and explores its behavior along segments in ample directions.

Contribution

It establishes the $C^{1,1}$ regularity of the volume function and analyzes its regularity properties in specific geometric directions.

Findings

Volume function is $C^{1,1}$ on the big cone.

Regularity along segments in ample directions is characterized.

Provides insights into the geometric behavior of volume functions.

Abstract

We prove the optimal $C^{1,1}$ regularity of the volume function on the big cone of a projective manifold, and investigate its regularity when restricted to segments moving in ample directions.