Geometric monodromy — semisimplicity and maximality

Abstract

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 $\pi_1(X,x)$ acting on the étale cohomology groups $H^*(Y_x,\mathbb{F}_\ell)$ are the reduction modulo-$\ell$ of those of $\pi_1(X,x)$ acting on $H^*(Y_x,\mathbb{Z}_\ell)$ 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 $\mathbb{F}_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture — namely, that $\pi_1(X,x)$ acts semisimply on $H^*(Y_x,\mathbb{F}_\ell)$ for $\ell\gg 0$ — is equivalent to the condition that the image of $\pi_1(X,x)$ acting on $H^*(Y_x,\mathbb{Q}_\ell)$ is `almost maximal’ (in a precise sense; what we call `almost hyperspecial’) with respect to the group of $\mathbb{Q}_\ell$-points of its Zariski closure. Ultimately, we prove the geometric variant with $\mathbb{F}_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture.