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Cycle-Level Intersection Theory for Toric Varieties

Published online by Cambridge University Press:  20 November 2018

Hugh Thomas
Hugh Thomas*
Affiliation:
The Fields Institute, 222 College St., Toronto, ON, M5T 3J1 e-mail: hthomas@fields.utoronto.ca
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Abstract

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This paper addresses the problem of constructing a cycle-level intersection theory for toric varieties. We show that by making one global choice, we can determine a cycle representative for the intersection of an equivariant Cartier divisor with an invariant cycle on a toric variety. For a toric variety defined by a fan in $N$, the choice consists of giving an inner product or a complete flag for ${{M}_{\mathbb{Q}}}\,=\,\mathbb{Q}\,\otimes \,\text{Hom(}N,\,\mathbb{Z}\text{)}$ , or more generally giving for each cone σ in the fan a linear subspace of ${{M}_{\mathbb{Q}}}$ complementary to ${{\sigma }^{\bot }}$ , satisfying certain compatibility conditions. We show that these intersection cycles have properties analogous to the usual intersections modulo rational equivalence. If $X$ is simplicial (for instance, if $X$ is non-singular), we obtain a commutative ring structure to the invariant cycles of $X$ with rational coefficients. This ring structure determines cycles representing certain characteristic classes of the toric variety. We also discuss how to define intersection cycles that require no choices, at the expense of increasing the size of the coefficient field.

Keywords

14M2514C17toric varietiesintersection theory
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 5 , 01 October 2004 , pp. 1094 - 1120
Copyright
Copyright © Canadian Mathematical Society 2004

References

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