# Schematic homotopy types and non-abelian Hodge theory

@article{Katzarkov2008SchematicHT,
title={Schematic homotopy types and non-abelian Hodge theory},
author={Ludmil Katzarkov and Tony Pantev and Bertrand Toen},
journal={Compositio Mathematica},
year={2008},
volume={144},
pages={582 - 632}
}
• Published 2008
• Mathematics
• Compositio Mathematica
Abstract We use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor$X \mapsto (X\otimes \mathbb {C})^{\mathrm {sch}}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on $(X\otimes \mathbb {C})^{\mathrm {sch}}$. This Hodge… Expand
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#### References

SHOWING 1-10 OF 93 REFERENCES
The Hodge filtration on nonabelian cohomology
This is partly a survey article on nonabelian Hodge theory, but we also give proofs of results that have only been announced elsewhere. In the introduction we discuss a wide range of recent work onExpand
Rational homotopy theory for non-simply connected spaces
• Mathematics
• 1999
We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends theExpand
F-isocrystals and homotopy types
Abstract We study a positive characteristic analogue of the nonabelian Hodge structure constructed by Katzarkov, Pantev, and Toen on the homotopy type of a complex algebraic variety. Given a properExpand
Algebras and Modules in Monoidal Model Categories
• Mathematics
• 1998
In recent years the theory of structured ring spectra (formerly known as A$_{\infty}$- and E$_{\infty}$-ring spectra) has been simplified by the discovery of categories of spectra with strictlyExpand
Champs affines
Abstract.The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraicExpand
Representative functions on discrete solvable groups
The topic here is the representation of discrete groups as automorphisms of finite-dimensional vector spaces over a field. The results here are mainly generalizations to the class of solvable groupsExpand
Local Projective Model Structures on Simplicial Presheaves
We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T . When T is the Nisnevich site it specializes to a proper simplicial model categoryExpand
The hodge de rham theory of relative malcev completion
Abstract Suppose that X is a smooth manifold and ρ : π1(X,x) → S is a representation of the fundamental group of X into a real reductive group with Zariski dense image. To such data one can associateExpand
On the derived category of perverse sheaves
Let D : Db(x, ~ ) be the usual derived category of ~-sheaves on a certain scheme X, and M = M{X)c D be the category of perverse sheaves for middle perversity. Now consider the derived category Db(M)Expand
MODEL CATEGORY STRUCTURES ON CHAIN COMPLEXES OF SHEAVES
In this paper, we try to determine when the derived category of an abelian category is the homotopy category of a model structure on the category of chain complexes. We prove that this is always theExpand