Filtrations on combinatorial intersection cohomology and invariants of subdivisions

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