Weighted Ehrhart theory via mixed Hodge modules on toric varieties

We give a cohomological and geometrical interpretation for the weighted Ehrhart theory of a full-dimensional lattice polytope P, with Laurent polynomial weights of geometric origin. For this purpose, we calculate the motivic Chern and Hirzebruch characteristic classes of a mixed Hodge module complex ℳ whose underlying cohomology sheaves are constant on the 𝕋-orbits of the toric variety X_P associated to P. Besides motivic coefficients, this also applies to the intersection cohomology Hodge module. We introduce a corresponding generalized Hodge χ_y-polynomial of the ample divisor D_P on X_P. Motivic properties of these characteristic classes are used to express this Hodge polynomial in terms of a very general weighed lattice point counting and the corresponding weighted Ehrhart theory. We introduce, for such a mixed Hodge modules complex ℳ on X, an Ehrhart polynomial E_P,ℳ generalizing the Hodge polynomial of ℳ and satisfying a reciprocity formula and a purity formula fitting with the duality for mixed Hodge modules. This Ehrhart polynomial and its properties depend only on a Laurent polynomial weight function on the faces Q of P. In the special case of the intersection cohomology mixed Hodge module, the weight function corresponds to Stanley's g-function of the polar polytope of P, hence it depends only on the combinatorics of P. In particular, we obtain a combinatorial formula for the intersection cohomology signature.