Cycles representing the Todd class of a toric variety

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 m(s) to each rational polyhedral cone s in the lattice, such that for any toric variety X with fan S, the Todd class of X is the sum over all cones s in S of m(s)[V(s)]. This constitutes an improved answer to an old question of Danilov. In a similar way, beginning with 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-named 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-named author.