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
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by James Pommersheim and Hugh Thomas

J. Amer. Math. Soc. 17 (2004), 983-994

DOI: https://doi.org/10.1090/S0894-0347-04-00460-6

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

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