Okounkov bodies and restricted volumes along very general curves

TL;DR

This paper demonstrates how the restricted volume of a big divisor along a very general curve can be derived from its Okounkov body, and shows that identical Okounkov bodies imply numerical equivalence of divisors.

Contribution

It establishes a method to compute restricted volumes from Okounkov bodies and proves that identical Okounkov bodies across all flags imply divisors are numerically equivalent.

Findings

Restricted volume can be read from the Okounkov body.

Equal Okounkov bodies across all flags imply numerical equivalence.

Provides a new link between geometric invariants and convex bodies.

Abstract

Given a big divisor $D$ on a normal complex projective variety $X$, we show that the restricted volume of $D$ along a very general complete-intersection curve $C\subset X$ can be read off from the Okounkov body of $D$ with respect to an admissible flag containing $C$. From this we deduce that if two big divisors $D_1$ and $D_2$ on $X$ have the same Okounkov body with respect to every admissible flag, then $D_1$ and $D_2$ are numerically equivalent.