  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.
