Filtrations on combinatorial intersection cohomology and invariants of subdivisions

Motivated by definitions in mixed Hodge theory, we define the weight filtration and the monodromy weight filtration on the combinatorial intersection cohomology of a fan. These filtrations give a natural definition of the multivariable invariants of subdivisions of polytopes, lattice polytopes and fans, namely the mixed $h$-polynomial, the refined limit mixed $h^*$-polynomial, and the mixed $cd$-index, defined by Katz--Stapledon and Dornian--Katz--Tsang. Previously, only the refined limit mixed $h^*$-polynomial had a geometric interpretation, which came from filtrations on the cohomology of a sch\"on hypersurface. Consequently, we generalize a positivity result on the mixed $h$-polynomial by Katz and Stapledon using the relative hard Lefschetz theorem of Karu.