ON THE SEMISIMPLICITY OF GEOMETRIC MONODROMY ACTION IN Fℓ-COEFFICIENTS

Let X be a smooth separated geometrically connected variety defined over a characteristic p finite field, f : Y → X a smooth projective morphism, and w a non-negative integer. A celebrated result of Deligne states that the higher direct image Qℓ-sheaf Rwf∗Qℓ is semisimple on X geometrically for all prime ℓ not equal to p. By comparing the invariant dimensions of sufficiently many ℓ-adic and mod ℓ representations arising from the sheaves Rwf∗Qℓ and Rwf∗Fℓ respectively, we prove that the Fℓ-sheaf Rwf∗Fℓ is likewise semisimple on X geometrically if ℓ is sufficiently large. As an application, a largeness result on the monodromy is obtained. This is a joint work with Anna Cadoret and Akio Tamagawa.