Intersection theory of toric 𝑏-divisors in toric varieties

Botero, Ana

doi:10.1090/jag/721

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

Author: Ana María Botero
Journal: J. Algebraic Geom. 28 (2019), 291-338
DOI: https://doi.org/10.1090/jag/721
Published electronically: January 10, 2019
MathSciNet review: 3912060
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Abstract | References | Additional Information

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.

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Additional Information

Ana María Botero
Affiliation: Institut für Mathematik, Technische Universität Darmstadt, Karolinenplatz 5, 64289 Darmstadt, Germany
Email: botero@mathematik.tu-darmstadt.de, anaboterocarrillo@gmail.com

Received by editor(s): April 5, 2017
Received by editor(s) in revised form: January 8, 2018
Published electronically: January 10, 2019
Additional Notes: This project was supported by the IRTG 1800 on Moduli and Automorphic Forms and by the Berlin Mathematical School.
Article copyright: © Copyright 2019 University Press, Inc.