Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications

Osserman, Brian

doi:10.2977/prims/1199403809

Cited by 15 publications

(16 citation statements)

References 19 publications

(23 reference statements)

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“…We prefer to give here a conceptual proof based on a geometric construction of the p-curvature due to Mochizuki and communicated to us by Brian Osserman; see [29]. This construction begins with the following crystalline interpretation of F * X/S Ω 1 X ′ /S .…”

Section: Proposition 15 the Action Described In Part (1) Of Propositmentioning

confidence: 99%

Nonabelian Hodge theory in characteristic p

2007

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Given a scheme in characteristic p together with a lifting modulo p 2 , we construct a functor from a category of suitably nilpotent modules with connection to the category of Higgs modules. We use this functor to generalize the decomposition theorem of Deligne-Illusie to the case of de Rham cohomology with coefficients. IntroductionLet X/C be a smooth projective scheme over the complex numbers and let X an be the associated analytic space. Classical Hodge theory provides a canonical isomorphism:Carlos Simpson's "nonabelian Hodge theory" [35] provides a generalization of this decomposition to the case of cohomology with coefficients in a representation of the fundamental group of X an . By the classical Riemann-Hilbert correspondence, such a representation can be viewed as a locally free sheaf E with integrable connection (E, ∇) on X. If (E, ∇) satisfies suitable conditions, Simpson associates to it a Higgs bundle (E ′ , θ), i.e., a locally free sheaf E ′ together with an O X -linear map θ :X/C vanishes. This integrability implies that the iterates of θ are zero, so that θ fits into a complex (the Higgs complex)As a substitute for the Hodge decomposition (0.0.1), Simpson constructs a natural isomorphism:In general, there is no simple relation between E and E ′ , and in fact the correspondence E → E ′ is not holomorphic. Our goal in this work is to suggest and investigate an analog of Simpson's theory for integrable connections in positive characteristics, as well as as an extension of the paper [8] of Deligne and Illusie to the case of de Rham cohomology with coefficients in a D-module. Let X be a smooth scheme over a the spectrum S of a perfect field k, and let F : X → X ′ be the relative Frobenius map. Assume as in [8] that there is a liftingX of X ′ to W 2 (k). Our main result is the construction of a functor CX (the Cartier transform) from the category MIC(X/S) of modules with integrable connection on X to the category HIG(X ′ /S) of Higgs modules on X ′ /S, each subject to suitable nilpotence conditions.The relative Frobenius morphism F and the p-curvatureof a module with integrable connection (E, ∇) play a crucial role in the study of connections in characteristic p. A connection ∇ on a sheaf of O X -modules E can be viewed as an action of the sheaf of PD-differential operators [3, (4.4)] 1 D X on X. This sheaf of rings has a large center Z X : in fact, F * Z X is canonically isomorphic to the sheaf of functions on the cotangent bundle T *is the pth iterate of θ and θ p is the pth power of θ in D X . If ∇ is an integrable connection on E, then by definition ψ θ is the O X -linear endomorphism of E given by the action of ∇ c(θ) .LetX be a lifting of X. Our construction of the Cartier transform CX is based on a study of the sheaf of liftings of the relative Frobenius morphismThe name "differential operators" is misleading: although D X acts on O X , the map′ is naturally a torsor under the group F * T X ′ . Key to our construction is the fact that the F * T X ′ -torsor q : LX → X has a canonical co...

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Section: Proposition 15 the Action Described In Part (1) Of Propositmentioning

confidence: 99%

Nonabelian Hodge theory in characteristic p

2007

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“…Let us denote by B the set of isomorphism classes of rank 2 semistable bundles V on X (1) such that det(V) ∼ = O X , and F * V is indecomposable and maximally unstable. Then it is well-known (cf., e.g., [32], § 4, p. 110, Proposition 4.2) that there is a natural 2 2g -to-1 correspondence between B and the set of isomorphism classes of dormant indigenous bundles on X/k. Thus, Corollary 5.4 of the present paper enables us to calculate the cardinality of B, i.e., to conclude that…”

Section: Relation With Other Resultsmentioning

confidence: 99%

An Explicit Formula for the Generic Number of Dormant Indigenous Bundles

Wakabayashi

2014

Publ. Res. Inst. Math. Sci.

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Abstract. A dormant indigenous bundle is an integrable P 1 -bundle on a proper hyperbolic curve of positive characteristic satisfying certain conditions. Dormant indigenous bundles were introduced and studied in the p-adic Teichmüller theory developed by S. Mochizuki. Kirti Joshi proposed a conjecture concerning an explicit formula for the degree over the moduli stack of curves of the moduli stack classifying dormant indigenous bundles. In this paper, we give a proof for this conjecture of Joshi.

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“…In the present note I describe a conjectural formula for this degree. For r = 2, g = 2, p ≥ 5 this was proved by [Mochizuki, 1999], [Osserman, 2007] and [Lange and Pauly, 2008] by completely different methods.…”

Section: Introductionmentioning

confidence: 93%

The Degree of the Dormant Operatic Locus

Joshi¹

2016

Int Math Res Notices

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Let X be a smooth, projective curve of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. I provide a conjectural formula for the degree of the scheme of dormant PGL(r)-opers on X where r ≥ 2 (I assume that p is greater than an explicit constant depending on g, r). For r = 2 a dormant PGL(2)-oper is a dormant indigenous bundle on X in the sense of Shinichi Mochuzki (and his work provides a formula only for g = 2, r = 2, p ≥ 5, from a different point of view). In 2014, Yasuhiro Wakabayashi has shown that my conjectural formula holds for r = 2, g ≥ 2 and p > 2g − 2 and more recently he has proved the conjecture in all ranks for generic curves of genus at least two. Hara-naka ya Mono nimo tsukazu Naku hibariMatsuo Basho [Miura, 1991]

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Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications

Cited by 15 publications

References 19 publications

Nonabelian Hodge theory in characteristic p

Nonabelian Hodge theory in characteristic p

An Explicit Formula for the Generic Number of Dormant Indigenous Bundles

The Degree of the Dormant Operatic Locus

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