几何单值性--半单纯性与极大性
Geometric monodromy -- semisimplicity and maximality
摘要：
设$X$是特征为$p\geq 0$的代数闭域$k$上光滑分离的连通概型，$f:Y\rightarrow X$是$X$上的光滑真态射，$x$是$X$上的几何点.证明了作用在上同调群$H^*(Y_x,F_\ell)$上的$π_1(X,x)$的有界长度$\leq d$的张量不变量是作用在$H^*(Y_x,Z_\ell)$上的$π_1(X,x)$的张量不变量的约化模-$\ell$，其中$\ell$大于一个只依赖于$f:Y\rightarrow X$，$d$的常数.我们应用这个结果来证明Grothendieck-Serre半单纯性猜想的具有$F_\ell$-系数的几何变式--即$π_1(X,x)$对$H^*(Y_x,F_\ell)$作用于$\ell\gg 0$的半单纯性--等价于$π_1(X,x)$作用于$H^*(Y_x,Q_\ell)$的象是"几乎极大"的条件（在精确的意义上，我们称之为"几乎超特殊"）关于它的Zapriki闭包的$Q_\ell$-点群。最后，我们证明了Grothendieck-Serre半单猜想的具有$F_\ell$-系数的几何变式.
英文摘要：
Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded length $\leq d$ of $π_1(X,x)$ acting on the étale cohomology groups $H^*(Y_x,F_\ell)$ are the reduction modulo-$\ell$ of those of $π_1(X,x)$ acting on $H^*(Y_x,Z_\ell)$ for $\ell$ greater than a constant depending only on $f:Y\rightarrow X$, $d$. We apply this result to show that the geometric variant with $F_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture -- namely that $π_1(X,x)$ acts semisimply on $H^*(Y_x,F_\ell)$ for $\ell\gg 0$ -- is equivalent to the condition that the image of $π_1(X,x)$ acting on $H^*(Y_x,Q_\ell)$ is `almost maximal' (in a precise sense; what we call `almost hyperspecial') with respect to the group of $Q_\ell$-points of its Zariski closure. Ultimately, we prove the geometric variant with $F_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture.
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