Cycles representing the Todd class of a toric variety

Pommersheim, James; Thomas, Hugh

doi:10.1090/S0894-0347-04-00460-6

Cycles representing the Todd class of a toric variety

Authors: James Pommersheim and Hugh Thomas
Journal: J. Amer. Math. Soc. 17 (2004), 983-994
MSC (2000): Primary 14M25; Secondary 14C17, 52B20
DOI: https://doi.org/10.1090/S0894-0347-04-00460-6
Published electronically: May 25, 2004
MathSciNet review: 2083474
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Abstract: In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product, we assign a rational number $\mu(\sigma)$ to each rational polyhedral cone $\sigma$ in the lattice, such that for any toric variety with fan $\Sigma$ in the lattice, we have

$\begin{displaymath}\operatorname{Td}(X)=\sum_{\sigma \in \Sigma} \mu(\sigma) [V(\sigma)].\end{displaymath}$

This constitutes an improved answer to an old question of Danilov.

In a similar way, beginning from the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli.

Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by the first author.

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Additional Information

James Pommersheim
Affiliation: Department of Mathematics, Pomona College, Claremont, California 92037
Email: jpommersheim@pomona.edu

Hugh Thomas
Affiliation: Fields Institute, 222 College Street, Toronto ON, M5T 3J1 Canada
Address at time of publication: Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3 Canada
Email: hthomas@fields.utoronto.ca

DOI: https://doi.org/10.1090/S0894-0347-04-00460-6
Keywords: Toric variety, Todd class, polytopes, counting lattice points
Received by editor(s): October 25, 2003
Published electronically: May 25, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.