Asymptotic cohomological functions of toric divisors

TL;DR

This paper investigates the growth rates of cohomology groups of divisors on toric varieties, revealing their piecewise polynomial nature and connecting geometric invariants to hyperplane arrangements.

Contribution

It introduces asymptotic cohomological functions for toric divisors, showing they are piecewise polynomial and linking divisor intersection numbers to hyperplane arrangement volumes.

Findings

Cohomological functions are piecewise polynomial.

Self-intersection numbers relate to hyperplane arrangement volumes.

Proves an asymptotic version of Serre vanishing.

Abstract

We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the self-intersection number of a T-Cartier divisor as a linear combination of the volumes of the bounded regions in the corresponding hyperplane arrangement and prove an asymptotic converse to Serre vanishing.