Weighted Ehrhart Theory via Mixed Hodge Modules on Toric Varieties

Abstract 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 ${\mathscr{M}}$ whose underlying cohomology sheaves are constant on the ${{\mathbb{T}}}$-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 $\chi _{y}$-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 ${\mathscr{M}}$ on $X_{P}$, an Ehrhart polynomial $E_{P,{\mathscr{M}}}$ generalizing the Hodge polynomial of ${\mathscr{M}}$ 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.