Intersection theory of toric 𝑏-divisors in toric varieties

Abstract

We introduce toric b b -divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions toric b 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 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 b -divisors with Newton–Okounkov bodies. The main motivation for studying toric b 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.