Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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

DOI: https://doi.org/10.1090/jag/721
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.