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Rigid connections and \(F\)-isocrystals. (English) Zbl 1491.14032

The primary objects to study in the paper are irreducible rigid rank \(r\) flat connections \((E, \nabla)\) with a torsion determinant line bundle \(L\), over a smooth projective variety \(X/\mathbb{C}\). C. T. Simpson [Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994; Zbl 0891.14005)] conjectured that such connections are of geometric origin, meaning that they are subquotients of Gauß-Manin connections of a family of smooth projective varieties defined on an open dense subvariety of \(X\). Moreover, it’s known that Gauß-Manin connections in characteristic \(p\) have nilpotent \(p\)-curvatures. So, Simpson’s conjecture predicts that mod-\(p\) reductions of rigid connections have nilpotent \(p\)-curvatures. The first main result of this paper confirms this expectation for \(p\) sufficiently large.
Based on the first result, the authors constructed \(F\)-isocrystalline realizations of such rigid connections. That is, they showed that there is a finite type \(\mathbb{Z}\)-scheme \(S\) over which \((X, (E, \nabla))\) has a model \((X_S, (E_S, \nabla_S))\), such that for all \(W(k)\)-points of \(S\), where \(k\) is a finite field, the \(p\)-adic completion of the base change \((\widehat{E}_{W(k)}, \widehat{\nabla}_{W(k)}))\) on \(\widehat{X}_{W(k)}\) defines a crystal on \(X_k/W(k)\) and the isocrystal \((\widehat{E}_{W(k)}, \widehat{\nabla}_{W(k)})) \otimes \mathbb{Q}\) is indeed an \(F\)-isocrystal after a base change to a finite field extension. There is a crystalline representation associated the constructed \(F\)-isocrystal, whose properties were also studied.
Proofs for those two main results rely on choices of arithmetic models \((X_S/S, L_S)\) of \((X/\mathbb{C}, L)\) (Lemma 3.1, Prop. 3.3, Prop. 4.10), and the fact that there are only finite number of isomorphism classes of rigid connections for fixed \(r\) and \(L\). An indispensable tool used in the proof of the second main theorem is the Higgs-de Rham flow developed in [G. Lan et al., J. Eur. Math. Soc. (JEMS) 21, No. 10, 3053–3112 (2019; Zbl 1444.14048)].
Finally, they showed that if there is an \(S\)-model \((X_S, (E_S, \nabla_S))\) of \((X, (E,\nabla))\) for a finite type \(\mathbb{Z}\)-scheme \(S\), satisfying that for each closed point \(s \in S\), \((E_s, \nabla_s)\) has vanishing \(p\)-curvature, then \((E, \nabla)\) has unitary monodromy. The proof is based on the study of the representation previously mentioned. This result can be seen as one step towards the understanding of the Grothendieck-Katz \(p\)-curvature conjecture.
A closely related paper to the one under review is [H. Esnault and M. Groechenig, Sel. Math., New Ser. 24, No. 5, 4279–4292 (2018; Zbl 1408.14037)], where another conjecture from [C. T. Simpson, Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994; Zbl 0891.14005)] was proved. Indeed, the result there was originally proved in a preprint version of the current one, proved as a consequence of the second main result here, together with the theory of \(p\)-to-\(\ell\)-companions [T. Abe and H. Esnault, Ann. Sci. Éc. Norm. Supér. (4) 52, No. 5, 1243–1264 (2019; Zbl 1440.14097)]. In [H. Esnault and M. Groechenig, Sel. Math., New Ser. 24, No. 5, 4279–4292 (2018; Zbl 1408.14037)], a shorter proof was given. Moreover, there the result was proved for quasi-project varieties.
There are still questions left open, e.g., if the first result holds for small primes \(p\).
Reviewer: Yun Hao (Berlin)

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14G22 Rigid analytic geometry
14H60 Vector bundles on curves and their moduli
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