Mixed Hodge structure in the cohomology of moment-angle complexes

Abstract

This work studies the relationship between the multigrading in the cohomology of moment-angle complexes, introduced by V. Buchstaber and T. Panov in their computation of these cohomology groups, and the multigrading in the cohomology of the corresponding complements of coordinate subspace arrangements, arising from the mixed Hodge structure. It is shown that these multigradings coincide. This result refines a theorem from Yu. Eliyashev's 2013 dissertation. Furthermore, results on the mixed Hodge structure in the cohomology of complements of coordinate subspace arrangements are applied to the study of the same varieties over finite fields. In particular, various formulas are obtained for the number of $\mathbb{F}_q$-rational points and for the associated global $ζ$-functions.