Regularity of the volume function

2025/05/20 by Valentino Tosatti, Cao, Junyu, Tosatti, Valentino
Engineering · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Enhanced Oil Recovery Techniques #FOS: Mathematics #Field-Flow Fractionation Techniques #Phase Equilibria and Thermodynamics

paper · pdf · doi:10.48550/arxiv.2505.14495

openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds
• Volume of Perturbations of Pseudoeffective Classes
• Notions of numerical Iitaka dimension do not coincide
• Duality between the pseudoeffective and the movable cone on a projective manifold
• Numerical analogues of the Kodaira dimension and the Abundance\n Conjecture
• Movable intersection and bigness criterion
• Comparing numerical dimensions
• On the birational automorphisms of varieties of general type
• Volume functions of linear series
• Monge-Ampère equations in big cohomology classes
• Convex Bodies Associated to Linear Series
• Differentiability of volumes of divisors and a problem of Teissier
• Restricted volumes and base loci of linear series
• Asymptotic invariants of line bundles
• The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
• Zariski chambers, volumes, and stable base loci
• Asymptotic invariants of base loci
• On the volume of a line bundle

Related

• Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds
• Boundary regularity of optimal transport maps on convex domains
• Characterizing volume via cone duality
• Volume of Perturbations of Pseudoeffective Classes
• Volume of line bundles via valuation vectors (different from Okounkov bodies)