2017/01/14 by Ana María Botero, Botero, Ana María
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1701.03925
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.