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Cycles representing the Todd class of a toric variety

Author(s): James Pommersheim; Hugh Thomas
Journal: J. Amer. Math. Soc. 17 (2004), 983-994.
MSC (2000): Primary 14M25; Secondary 14C17, 52B20
Posted: May 25, 2004
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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 $X$ 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.

References:

[BP]
A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, New Perspectives in Algebraic Combinatorics, MSRI Publications 38, 1999, pp. 91-147. MR 1731815 (2000k:52014)

[Dan]
V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33:2 (1978), 97-154. MR 0495499 (80g:14001)

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W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies Number 131, Princeton University Press, Princeton, 1993.MR 1234037 (94g:14028)

[McM]
P. McMullen, Weakly continuous valuations on convex polytopes, Archiv Math. 41 (1983) 555-564. MR 0731639 (85i:52002)

[Mor]
R. Morelli, Pick's theorem and the Todd class of a toric variety, Adv. Math. 100:2 (1993), 183-231. MR 1234309 (94j:14048)

[Pom1]
J. E. Pommersheim, Toric varieties, lattice points and Dedekind sums, Math. Ann. 295 (1993) 1-24. MR 1198839 (94c:14043)

[Pom2]
J. E. Pommersheim, Products of Cycles and the Todd Class of a toric variety, J. Amer. Math. Soc. 9 (1996) 813-826. MR 1358042 (96j:14037)

[Tho]
H. Thomas, Cycle-level intersection theory for toric varieties, to appear in Canad. J. Math., available at http://www.arxiv.org/math.AG/0306144.

[Wlo]
J. W\lodarczyk, Decomposition of birational toric maps in blow-ups and blow-downs, Trans. Amer. Math. Soc. 349 (1997), 373-411. MR 1370654 (97d:14021)

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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: 10.1090/S0894-0347-04-00460-6
PII: S 0894-0347(04)00460-6
Keywords: Toric variety, Todd class, polytopes, counting lattice points
Received by editor(s): October 25, 2003
Posted: May 25, 2004
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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