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Description

We prove that the monodromy of a cohomologically rigid integrable connection $(E,\nabla)$ on a smooth complex projective variety $X$ is integral. This answers positively a special case of a conjecture by Carlos Simpson. To this aim, we prove that the mod $p$ reduction of a rigid integrable connection $(E,\nabla) $ has the structure of an isocrystal with Frobenius structure. We also prove that rigid integrable connections with vanishing $p$- curvatures are unitary. This allows one to prove new cases of Grothendieck’s $p$-curvature conjecture. Joint with Michael Groechenig