Nonabelian Hodge theory in characteristic \(p\)
Let \(X/\mathbb{C}\) be a smooth projective scheme over the field of complex numbers and let \(X^{an}\) be the associated analytic space. Classical Hodge theory provides a canonical isomorphism \[ H^n(X, \Omega^\bullet_{X/\mathbb{C}})\cong H^n(X^{an},\mathbb{C})\cong \bigoplus_{i+j= n}H^i(X, \Omega^j_{X/\mathbb{C}}), \] which is generally known as the classical Hedge decomposition theorem. About 15 years ago, \textit{C. T. Simpson} [Publ. Math., Inst. Hautes Étud. Sci. 75, 5--95 (1992; Zbl 0814.32003)] generalized this decomposition to the case of cohomology with coefficients in a representation of the fundamental group of the analytic space \(X^{an}\), which led to what he called ``nonabelian Hodge theory''. More precisely, any representation of the fundamental group of \(X^{an}\) can be viewed as a locally free sheaf \(E\) with integrable connection \((E,\nabla)\) on \(X\), basically via the classical Riemann-Hilbert correspondence. If \((E,\nabla)\) satisfies suitable conditions, then Simpson's construction associates to it a so-called Higgs bundle \((E',\theta)\), together with a complex \[ E'\otimes \Omega^\bullet_{X/\mathbb{C}}:= E'\to E'\otimes \Omega^1_{X/\mathbb{C}}\to E'\otimes \Omega^2_{X/\mathbb{C}}\to \cdots \] and natural isomorphisms \[ H^n(X, E\otimes \Omega^\bullet_{X/\mathbb{C}}, d)\cong H^n(X^{an}, V)\cong H^n(X, E'\otimes \Omega^\bullet_{X/\mathbb{C}},\theta), \] where \(V\) is the respective representation space of the fundamental group of \(X^{an}\). The main goal of the comprehensive treatise under review is to suggest and to develop an analog of Simpson's theory for integrable connections in characteristic \(p> 0\), as well as an extension of earlier results of \textit{P. Deligne} and \textit{L. Illusie} [Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)] to the case of de Rham cohomology in positive characteristics for sheaves of modules with an integrable connection. To this end, the authors consider an arbitrary smooth scheme \(X\) over the spectrum \(S\) of a perfect field \(k\) and the associated relative Frobenius map \(F: X\to X'\). Assume as in the approach of Deligne and Illusie that there is a lifting \(\widetilde X\) of \(X'\) to the Witt ring \(W_2(k)\). Their main result is then the construction of a functor \(C_{\widetilde X}\), the so-called Cartier transform, from the category \(\text{MIC}(X/S)\) of sheaves of modules with integrable connection on \(X\) to the category \(\text{HIG}(X'/S)\) of Higgs modules on \(X'/S\) each of which is subject to certain nilpotence conditions. The construction of the Cartier transform is based on a refined study of the sheaf of liftings of the relative Frobenius morphism \(F: X\to X'\) with respect to \(\widetilde X\) which appears to be a \(F^* T_{X'}\)-torsor over \(X\) admitting a canonical connection. The subtle construction of the Cartier transform, which involves elaborate techniques from the theory of \({\mathcal D}\)-modules, curvature of connections, and Azumaya algebras in characteristic \(p> 0\) is carried out in the first three chapters of the current paper. Chapter 1 introduces the torsor of Frobenius liftings, whereas Chapter 2 provides the actual construction of the Cartier transform in the context of Azumaya algebras, their étale splittings, the Riemann-Hilbert correspondence, and the de Rham cohomology in positive characteristics. Chapter 3 is devoted to the functoriality of the Cartier transform, thereby using. Gauss-Mann connections and Higgs fields, de Rham direct images, derived direct and inverse images, and suitable filtrations. The remainder of this chapter provides derived versions of the foregoing results in a certain filtered derived category of \({\mathcal D}_{X/S}\)-modules in characteristic \(p> 0\) which generalize the previous related work by \textit{G. Laumon} [Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982, Lect. Notes Math. 1016, 151--237 (1983; Zbl 0551.14006)] and by \textit{A. Braverman} and \textit{R. Bezrukavnikov} [Pure Appl. Math. Q. 3, No. 1, 153--179 (2007; Zbl 1206.14030)]. Chapter 4 is devoted to various applications and concrete examples of the authors' general approach, including a local study of the \(p\)-curvature, stacks of liftings and splittings, line bundles with connection, Abelian varieties in characteristic \(p> 0\), Fontaine modules, and a proof by reduction modulo \(p\) of a celebrated recent theorem of Barannikov and Kontsevich as published by \textit{C. Sabbah} [Tohoku Math. J., II. Ser. 51, No. 1, 125--140 (1999; Zbl 0947.14007)]. The authors conclude their highly substantial and propelling treatise with an appendix on generalities about Higgs fields, Higgs transforms, Higgs fields over group schemes and their convolutions, Azumaya algebras over group schemes, and the allied framework of tensor categories. Most of this material is treated systematically for the first time, with full proofs and explicit examples provided throughout.
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