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.
Citation
Shoji Yokura. "Local comparisons of homological and homotopical mixed Hodge polynomials." Proc. Japan Acad. Ser. A Math. Sci. 96 (3) 28 - 31, March 2020. https://doi.org/10.3792/pjaa.96.006
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