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D-modules and finite monodromy

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We investigate an analogue of the Grothendieck p-curvature conjecture, where the vanishing of the p-curvature is replaced by the stronger condition, that the module with connection mod p underlies a \({{\mathcal {D}}}_X\)-module structure. We show that this weaker conjecture holds in various situations, for example if the underlying vector bundle is finite in the sense of Nori, or if the connection underlies a \({{\mathbb {Z}}}\)-variation of Hodge structure. We also show isotriviality assuming a coprimality condition on certain mod p Tannakian fundamental groups, which in particular resolves in the projective case a conjecture of Matzat–van der Put.

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We thank Yves André, Antoine Chambert-Loir and Johan de Jong for their interest and for discussions. We especially thank Sinan Ünver for a close, and perceptive reading of the manuscript, and João Pedro dos Santos for mentioning [21] to us. We thank the referee for a thorough reading and useful comments. The first named author thanks the department of mathematics of Harvard University for hospitality during the preparation of this work.

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Correspondence to Hélène Esnault.

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To Alexander Beilinson, on the occasion of his 60th birthday, with admiration

The first author is supported by the Einstein program. The second author was partially supported by NSF Grant DMS-0017749000.

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Esnault, H., Kisin, M. D-modules and finite monodromy. Sel. Math. New Ser. 24, 145–155 (2018).

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