Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting
Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting
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通过点计数配置空间和最大环面空间的扭曲上同调
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发表时间：
2016
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通讯作者：
Weiyan Chen
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作者：
Weiyan Chen
本文考虑两类以自然数$n$为指标的代数簇$Y_n$：$mathbb{C}$上不同点的无序$n$-元组的构形空间和$mathbb{C}^n$上线性无关线的无序$n$-元组的空间。设W_n是由特征多项式给出的任意虚S_n-表示序列，对所有i和所有n，用二重生成函数计算H^i（Y_n; W_n）。计算的结果之一是一个新的递归现象：稳定的扭曲Betti数$lim_{n {0}个dim H^i（Y_n; W_n）$在i$中线性常返.我们的方法是计算某些代数簇的$F_q$-点上的扭点数，然后通过Grothendieck-Lefschetz不动点公式在拓扑学上证明结果。我们还推广了Church-Ellenberg-Farb关于仿射线的位形空间的一个结果到一般光滑簇的位形空间。
We consider two families of algebraic varieties $Y_n$ indexed by natural numbers $n$: the configuration space of unordered $n$-tuples of distinct points on $mathbb{C}$, and the space of unordered $n$-tuples of linearly independent lines in $mathbb{C}^n$. Let $W_n$ be any sequence of virtual $S_n$-representations given by a character polynomial, we compute $H^i(Y_n; W_n)$ for all $i$ and all $n$ in terms of double generating functions. One consequence of the computation is a new recurrence phenomenon: the stable twisted Betti numbers $lim_{n oinfty}dim H^i(Y_n; W_n)$ are linearly recurrent in $i$. Our method is to compute twisted point-counts on the $F_q$-points of certain algebraic varieties, and then pass through the Grothendieck-Lefschetz fixed point formula to prove results in topology. We also generalize a result of Church-Ellenberg-Farb about the configuration spaces of the affine line to those of a general smooth variety.