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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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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