Nonabelian Hodge theory in characteristic \(p\)
type: Item
instance of: scholarly article
title: Nonabelian Hodge theory in characteristic \(p\)
zbMATH Open document ID: 1140.14007
author: Elsie Sterbin Gottlieb
published in: Publications Mathématiques
publication date: 2008-05-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0507476
full work available at URL: https://eudml.org/doc/104228
review text: 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 th…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14F43
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14K15
zbMATH DE Number: 5277424
zbMATH Keywords: Hodge theory
zbMATH Keywords: non-Abelian theory
zbMATH Keywords: Higgs bundles
zbMATH Keywords: Higgs field
zbMATH Keywords: de Rham cohomology
zbMATH Keywords: Higgs cohomology
zbMATH Keywords: Riemann-Hilbert correspondence
zbMATH Keywords: Azumaya algebras
MaRDI profile type: Publication
OpenAlex ID: W2094512026
arXiv ID: math/0507476
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DOI: 10.1007/S10240-007-0010-Z
description: scientific article