Local comparisons of homological and homotopical mixed Hodge polynomials

Abstract

For a simply connected complex algebraic variety $X$, by the mixed Hodge structures $(W_{\bullet}, F^{\bullet})$ and $(\tilde{W}_{\bullet}, \tilde{F}^{\bullet})$ of the homology group $H_{*}(X;\mathbf{Q})$ and the homotopy groups $\pi_{*}(X)\otimes \mathbf{Q}$ respectively, we have the following mixed Hodge polynomials \begin{equation*} \mathit{MH}_{X}(t,u,v):= ∑_{k,p,q} \dim (\mathit{Gr}_{F_{•}}^{p} \mathit{Gr}^{W_{•}}_{p+q} H_{k} (X;\mathbf{C})) t^{k} u^{-p} v^{-q}, \end{equation*} \begin{equation*} \mathit{MH}^{π}_{X}(t,u,v):= ∑_{k,p,q} \dim (\mathit{Gr}_{\tilde{F}_{•}}^{p} \mathit{Gr}^{\tilde{W}_{•}}_{p+q} (π_{k}(X) øtimes \mathbf{C})) t^{k}u^{-p} v^{-q}, \end{equation*} which are respectively called the homological mixed Hodge polynomial and the homotopical mixed Hodge polynomial. In this paper we discuss some inequalities concerning these two mixed Hodge polynomials.