Weighted Ehrhart Theory via Mixed Hodge Modules on Toric Varieties.

The article focuses on providing a cohomological and geometric interpretation of weighted Ehrhart theory for full-dimensional lattice polytopes, using Laurent polynomial weights derived from mixed Hodge module complexes on the associated projective toric variety. It establishes that weighted Ehrhart polynomials, defined via these weights on the faces of the polytope, correspond to generalized Hodge \(\chi_y\)-polynomials computed through Hirzebruch characteristic classes and satisfy reciprocity and purity formulas linked to duality in mixed Hodge modules. In particular, when the mixed Hodge module is the intersection cohomology complex, the weights coincide with Stanley’s \(g\)-polynomials of the polar polytope, yielding combinatorial formulas for intersection cohomology signatures and extending classical Ehrhart reciprocity. The paper develops these results through the framework of motivic Chern and Hirzebruch classes, stratifications by torus orbits, and a generalized Hirzebruch–Riemann–Roch theorem, providing explicit formulas and applications in toric geometry.