2022/05/09 by Tsang, Ling Hei
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.03992
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 schön hypersurface. Consequently, we generalize a positivity result on the mixed h-polynomial by Katz and Stapledon using the relative hard Lefschetz theorem of Karu.