论文标题
同源和同型混合霍奇多项式的局部比较
Local comparisons of homological and homotopical mixed Hodge polynomials
论文作者
论文摘要
对于简单连接的复杂代数$ x $，由混合杂货结构$（w _ {\ bullet}，f^{\ bulter}）$和$（\ tilde w _ {\ bullet}，\ tilde f^{\ buttral}）$ homology $ homology $ h $ h _ _ _} $π_ {*}（x）\ otimes \ mathbb q $，我们有以下混合混合hodge polyenmials $$ mh_x（t，u，u，v）：= \ sum_ {k，p，p，q} \ perperatOrnAme {dim} {dim} gr^{w _ {\ bullet}} _ {p+q} h_k（x; \ mathbb c）\ bigr）t^{k} u^{ - p} v^{ - q}，$ $ $ $ $ $ $ $ $ $ \ quad \ quad \ quad \ quad \，\，\，\，mh^^^^fim_x^π_x（mh^π_x（c） \ operatorName {dim} \ bigl（gr _ {\ tilde f^{\ bullet}}}}^{p} gr^{\ tilde w _ {\ bulter}} _ {p+q} v^{ - q}，$$分别称为\ emph {同源混合霍奇多项式}和\ emph {同型混合混合霍奇多项式}。在本文中，我们讨论了有关这两个混合杂货多项式的一些不平等。
For a simply connected complex algebraic variey $X$, by the mixed Hodge structures $(W_{\bullet}, F^{\bullet})$ and $(\tilde W_{\bullet}, \tilde F^{\bullet})$ of the homology group $H_{*}(X;\mathbb Q)$ and the homotopy groups $π_{*}(X)\otimes \mathbb Q$ respectively, we have the following mixed Hodge polynomials $$MH_X(t,u,v):= \sum_{k,p,q} \operatorname{dim} \Bigl ( Gr_{F^{\bullet}}^{p} Gr^{W_{\bullet}}_{p+q} H_k (X;\mathbb C) \Bigr) t^{k} u^{-p} v^{-q},$$ $$\quad \, \, MH^π_X(t,u,v):= \sum_{k,p,q} \operatorname{dim} \Bigl (Gr_{\tilde F^{\bullet}}^{p} Gr^{\tilde W_{\bullet}}_{p+q} (π_k(X) \otimes \mathbb C) \Bigr ) t^ku^{-p} v^{-q},$$ which are respectively called \emph{the homological mixed Hodge polynomial} and \emph{the homotopical mixed Hodge polynomial}. In this paper we discuss some inequalities concerning these two mixed Hodge polynomials.
