Cycle-Level Products in Equivariant Cohomology of Toric Varieties

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.Comment: 15 pages, to be published in Michigan Mathematical Journal; LaTe

Barvinok's algorithm and the Todd class of a toric variety

AbstractIn this paper we prove that the Todd class of a simplicial toric variety has a canonical expression as a power series in the torus-invariant divisors. Given a resolution of singularities corresponding to a nonsingular subdivision of the fan, we give an explicit formula for this power series which yields the Todd class. The computational feasibility of this procedure is implied by the additional fact that the above formula is compatible with Barvinok decompositions (virtual subdivisions) of the cones in the fan. In particular, this gives an algorithm for determining the coefficients of the Todd class in polynomial time for fixed dimension. We use this to give a polynomial-time algorithm for computing the number of lattice points in a simple lattice polytope of fixed dimension, a result first achieved by Barvinok

An Algorithmic Theory of Lattice Points in Polyhedra

We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higher-dimensional Dedekind sums, complexity of the Presburger arithmetic, efficient computations with rational functions, and others. Although the main slant is algorithmic, structural results are discussed, such as relations to the general theory of valuations on polyhedra and connections with the theory of toric varieties. The paper surveys known results and presents some new results and connections