Regularity of the volume function

TL;DR

The paper proves that the volume function $Vol$ is $C^{1,1}_{\rm loc}$ on the big cone $\mathcal{B}_X$ for projective (or BDPP) manifolds and that $Vol$ is locally Lipschitz on all of $H^{1,1}(X,\mathbb{R})$. It provides two independent proofs of the $C^{1,1}$ regularity, one via a concavity approach with Fujita approximations and Khovanskii-Teissier inequalities and another via a general Lipschitz criterion for homogeneous, non-decreasing functions on a convex cone. The work also shows that this level of regularity is optimal along certain segments: there exist examples where $t\mapsto Vol(\alpha+t\omega)$ fails to be $C^{2,\gamma}$ on small intervals, highlighting intrinsic limits to smoothness. The results offer insight into the differentiability structure of the volume function and connect to broader questions about wall-chamber decompositions and a.e. differentiability.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Proposition 2.1
• proof
• proof : First proof of Theorem \ref{['t1']}