Geometric monodromy - Semisimplicity and maximality

Abstract

© 2017 Department of Mathematics, Princeton University. Let X be a connected scheme, smooth and separated over an alge- braically closed field k of characteristic p ≥ 0, let f: Y → X be a smooth proper morphism and x a geometric point on X. We prove that the tensor invariants of bounded length ≤ d of π1(X; x) acting on the étale cohomology groups H*(Yx; Fℓ) are the reduction modulo-ℓ of those of π1(X, x) acting on H*(Yx, ℤℓ) for ℓ greater than a constant depending only on f: Y → X, d. We apply this result to show that the geometric variant with Fℓ-coefficients of the Grothendieck-Serre semisimplicity conjecture - namely, that π1(X, x) acts semisimply on H*(Yx, Fℓ) for ℓ ≫ 0-is equivalent to the condition that the image of π1(X, x) acting on H*(Yx;Qℓ) is 'almost maximal' (in a precise sense; what we call 'almost hyperspecial') with respect to the group of Qℓ-points of its Zariski closure. Ultimately, we prove the geometric variant with Fℓ-coefficients of the Grothendieck-Serre semisimplicity conjecture.