## Cycles representing the Todd class of a toric variety

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- by James Pommersheim and Hugh Thomas PDF
- J. Amer. Math. Soc.
**17**(2004), 983-994 Request permission

## 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 \[ \operatorname {Td}(X)=\sum _{\sigma \in \Sigma } \mu (\sigma ) [V(\sigma )].\] 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

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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
- MR Author ID: 649257
- ORCID: 0000-0003-1177-9972
- Email: hthomas@fields.utoronto.ca
- Received by editor(s): October 25, 2003
- Published electronically: May 25, 2004
- © Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2083474