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...
To Joseph Bernstein with love and gratitude.Let Var be the category of complex algebraic varieties, Top that of nice topological spaces, D ab be the derived category of finite complexes of finitely generated abelian groups. One has tensor functors Var → Top → D ab , the first assigns to a variety its space equipped with the classical topology, the second one is the singular chain complex functor (the tensor structure for the first two categories is given by the direct product). The basic objective of the motive theory is to fill in a commutative squarewhere D M -the category of motives -is a rigid tensor triangulated category defined, together with the upper horizontal arrow, in a purely geometric way (so that the base field C can be replaced by any field), and the right vertical arrow is a tensor triangulated functor (which absorbs all the transcendence of the singular chains).The known constructions (due to Hanamura, Levine, and Voevodsky) proceed by first embedding Var into a larger DG category, and then define D M as its appropriate quotient. Voevodsky's generators and relations are especially neat. A rough idea: consider the localization of the DG category freely generated by the category of topological manifolds modulo the relations that kill the complexes of typesis an open covering of X); the singular chains functor yields then an equivalence between this toy category of "topological motives" and D ab . To define D M , one formally imitates this construction in the algebro-geometric setting with an important modification: mere combinations of true algebraic maps should be replaced from scratch by a larger group of finite correspondences, i.e., multi-valued maps (which is irrelevant in the topological setting).What follows is a concise exposition of Voevodsky's theory that covers principal points of and [MVW] with the notable exception of comparison results (relating the motivic cohomology with Bloch's higher Chow groups and Milnor's Kgroups, and theétale localized motives with finite coefficients with Galois modules, see [MVW] 19.1, 5.1, [Vo2] 3.3.3). The more advanced subjects of A 1 -homotopy theory, the proof of the Milnor-Bloch-Kato conjecture, and the array of motivic dreams, are not touched.We take the time to spell out the basic constructions on the DG category level ([Vo2] and [MVW] consider mere triangulated category structure). For the present material, this has the limited advantage of making formulas like (4.4.1), (4.4.2) 1
We study integrality of instanton numbers (genus zero Gopakumar - Vafa invariants) for quintic and other Calabi-Yau manifolds. We start with the analysis of the case when the moduli space of complex structures is one-dimensional; later we show that our methods can be used to prove integrality in general case. We give an expression of instanton numbers in terms of Frobenius map on $p$-adic cohomology ; the proof of integrality is based on this expression.Comment: 10 pages, minor change
with an Appendix by Vadim Vologodsky To the memory of Andrei ZelevinskyMark Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product S n (Z/rZ) n . He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of A 2n by the symmetric group S n .A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and Kaledin [2] via quantization in positive characteristic. In the present note we prove various properties of these derived equivalences and then deduce generalized Macdonald positivity for wreath products.
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