Rigid connections and $F$-isocrystals

Abstract: An irreducible integrable connection $(E,\nabla)$ on a smooth projective complex variety $X$ is called rigid if it gives rise to an isolated point of the corresponding moduli space $\mathcal{M}<em>{dR}(X)$. According to Simpson's motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gau\ss-Manin connection of a family of smooth projective varieties defined on an open dense subvariety of $X$. In this article we study mod $p$ reductions of irreducible rigid connections and establish results which confirm Simpson's prediction. In particular, for large $p$, we prove that $p$-curvatures of mod $p$ reductions of irreducible rigid flat connections are nilpotent, and building on this result, we construct an $F$-isocrystalline realization for {irreducible} rigid flat connections. More precisely, we prove that there exist smooth models $X_R$ and $(E_R,\nabla_R)$ of $X$ and $(E,\nabla)$, over a finite type ring $R$, such that for every Witt ring $W(k)$ of a finite field $k$ and every homomorphism $R \to W(k)$, the $p$-adic completion of the base change $$(\widehat{E}</em>{W(k)},\widehat{\nabla}<em>{W(k)})$$ on $\widehat{X}</em>{W(k)}$ represents an $F$-isocrystal. Subsequently we show that {irreducible} rigid flat connections with vanishing $p$-curvatures are unitary. This allows us to prove new cases of the Grothendieck--Katz $p$-curvature conjecture. We also prove the existence of a complete companion correspondence for $F$-isocrystals stemming from irreducible cohomologically rigid connections.