Nonabelian Hodge theory in characteristic p

Abstract

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...

Singular Localization and Intertwining Functors for Reductive Lie Algebras in Prime Characteristic

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“…The equivalence of (1) and (5) is the original result of Deligne and Illusie [8]. The implication (1) ⇒ (2) was proven by Ogus and Vologodsky [10], where a splitting module is explicitly constructed. The implications (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (2) are parts of Theorem 3.7, and give a geometric interpretation of Deligne and Illusie's result.…”

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“…An algebraic proof of Bogomolov's inequality for Higgs sheaves became possible only after the appearance of Ogus and Vologodsky's non-abelian Hodge theory in positive characteristic (see [31]). We also use in a crucial way the results of [21] to prove a weak version of Bogomolov's inequality for semistable modules with generalized connections.…”