Products of Cycles and the Todd Class of a Toric Variety

The purpose of this paper is to show that the Todd class of a simplicial toric variety has a canonical expression in terms of products of torus-invariant divisors. The coefficients in this expression, which are generalizations of the classical Dedekind sum, are shown to satisfy a reciprocity relation which characterizes them uniquely. We achieve these results by giving an explicit formula for the push-forward of a product of cycles under a proper birational map of simplicial toric varieties. Since the introduction of toric varieties in the 1970s, finding formulas for their Todd class has been an interesting and important problem. This is partly due to a well-known application of the Riemann-Roch theorem which allows a formula for the Todd class of a toric variety to be translated directly into a formula for the number of lattice points in a lattice polytope (cf. [Dan]). An example of this application is contained in [Pom], where a formula for the Todd class of a toric variety in terms of Dedekind sums is used to obtain new lattice point formulas. Danilov [Dan] posed the problem of finding a formula for the Todd class of a toric variety in terms of the orbit closures under the torus action. Specifically, he asked if it is possible, given a lattice, to assign a rational number to each cone in the lattice such that given any fan in the lattice, the Todd class of the corresponding toric variety equals the sum of the orbit closures with coefficients given by these assigned rational numbers. Morelli [Mor] showed that such an assignment is indeed possible in a natural way if the coefficients, instead of being rational numbers, are allowed to take values in the field of rational functions on a Grassmannian of linear subspaces of the lattice. However, if it is required that the coefficients be rational numbers invariant under lattice automorphisms, such an assignment is clearly impossible. For example, the nonsingular cone σ in Z generated by (1, 0) and (0, 1) when subdivided by the ray through (1, 1) yields two cones σ1 and σ2 which are both lattice equivalent to σ. By additivity, a consequence of the fact that the Todd class pushes forward, we deduce that the coefficient assigned to σ must equal 0, which is absurd. In this paper, we show that there is a canonical expression for the Todd class of a simplicial toric variety in terms of products of the torus invariant divisors. Furthermore, this expression is invariant under lattice automorphisms. That is, the coefficient of each product depends only on the set of rays with multiplicities