Ehrhart theory for Lawrence polytopes and orbifold cohomology of hypertoric varieties

Abstract

We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are equal to the coefficients of the Ehrhart δ \delta -polynomial of the associated Lawrence polytope. As a consequence, we deduce a formula for the Ehrhart δ \delta -polynomial of a Lawrence polytope and use the injective part of the Hard Lefschetz Theorem for hypertoric varieties to deduce some inequalities between the coefficients of the δ \delta -polynomial.