Intersection theory of toric $b$-divisors in toric varieties

TL;DR

This paper develops a theory of toric b-divisors on smooth toric varieties, establishing their integrability, computing degrees via convex volumes, and relating their global sections to lattice points, thus generalizing classical toric divisor results.

Contribution

It introduces the concept of toric b-divisors, proves their integrability under positivity assumptions, and connects their properties to convex geometry and Newton--Okounkov bodies.

Findings

Toric b-divisors are integrable under positivity conditions.

Degree of a toric b-divisor equals the volume of an associated convex set.

Dimension of global sections equals the lattice point count in the convex set.

Abstract

We introduce toric $b$-divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions toric $b$-divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric $b$-divisor is equal to the number of lattice points in this convex set and we give a Hilbert--Samuel type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. Finally, we relate convex bodies associated to $b$-divisors with Newton--Okounkov bodies. The main motivation for studying toric $b$-divisors is that they locally encode the singularities of the invariant metric on an automorphic line bundle over a toroidal compactification of a mixed Shimura variety of non-compact type.