Cycle-Level Products in Equivariant Cohomology of Toric Varieties

TL;DR

This paper introduces a new way to represent equivariant Todd classes of toric varieties using cycle-level products, extending existing theories and connecting to combinatorial formulas.

Contribution

It defines an action of equivariant Cartier divisors on cycle groups, lifting cohomology multiplication to cycles and unifying various Todd class constructions.

Findings

Cycle-level product action on equivariant cycles

Equivariant Todd class represented by cycle classes

Connection to local Euler-Maclaurin formula

Abstract

In this paper, we define an action of the group of equivariant Cartier divisors on a toric variety X on the equivariant cycle groups of X, arising naturally from a choice of complement map on the underlying lattice. If X is nonsingular, this gives a lifting of the multiplication in equivariant cohomology to the level of equivariant cycles. As a consequence, one naturally obtains an equivariant cycle representative of the equivariant Todd class of any toric variety. These results extend to equivariant cohomology the results of Thomas and Pommersheim. In the case of a complement map arising from an inner product, we show that the equivariant cycle Todd class obtained from our construction is identical to the result of the inductive, combinatorial construction of Berline-Vergne. In the case of arbitrary complement maps, we show that our Todd class formula yields the local Euler-Maclarurin formula introduced in Garoufalidis-Pommersheim.