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September 2020 Rigid connections and $F$-isocrystals
Hélène Esnault, Michael Groechenig
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Acta Math. 225(1): 103-158 (September 2020). DOI: 10.4310/ACTA.2020.v225.n1.a2


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 $\mathcal{M}_{\rm dR}(X)$. According to Simpson’s motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gauss–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 $(\widehat{E}_{W(k)},\widehat{\nabla}_{W(k)})$ on $\widehat{X}_{W(k)}$ 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.

Funding Statement

The first author was supported by the Einstein program and the ERC Advanced Grant 226257, the second author was supported by a Marie Skłodowska–Curie fellowship. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie Grant Agreement No. 701679.


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Hélène Esnault. Michael Groechenig. "Rigid connections and $F$-isocrystals." Acta Math. 225 (1) 103 - 158, September 2020.


Received: 8 February 2018; Revised: 20 August 2019; Published: September 2020
First available in Project Euclid: 16 January 2021

Digital Object Identifier: 10.4310/ACTA.2020.v225.n1.a2

Rights: Copyright © 2020 Institut Mittag-Leffler


Vol.225 • No. 1 • September 2020
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