Geometric monodromy — semisimplicity and maximality

Anna Cadoret¹ ; Chun-Yin Hui² ; Akio Tamagawa³

¹ Centre de Mathématiques Laurent Schwartz (UMR 7640), Ecole Polytechnique, 91128 Palaiseau, France
² VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
³ Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Annals of mathematics, Tome 186 (2017) no. 1, pp. 205-236.

Résumé

Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\ge 0$, let $f:Y\to X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded length $\le d$ of ${\pi }_1(X,x)$ acting on the étale cohomology groups $H^{\ast }(Y_x,{\mathbb{F}}_{\ell })$ are the reduction modulo-$\ell$ of those of ${\pi }_1(X,x)$ acting on $H^{\ast }(Y_x,{\mathbb{Z}}_{\ell })$ for $\ell$ greater than a constant depending only on $f:Y\to X$, $d$. 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 $H^{\ast }(Y_x,{\mathbb{F}}_{\ell })$ for $\ell \gg 0$ — is equivalent to the condition that the image of ${\pi }_1(X,x)$ acting on $H^{\ast }(Y_x,{\mathbb{Q}}_{\ell })$ 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.

Détail

DOI : 10.4007/annals.2017.186.1.5