# 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} }

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

#### 29 Citations

Schematization of homotopy types and realizations

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1Introduction Using ground breaking results of Simpson in ajoint work with T. Pantev and B.Toen [KTP] we have found new homotopy invariants of atopological space $X$ , related to the action of… Expand

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The purpose of this paper is to generalise Sullivan's rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toen's schematic homotopy types over any field k… Expand

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We show that if X is any proper complex variety, there is a weight decomposition on the real schematic homotopy type, in the form of an algebraic G_m-action. This extends to a real Hodge structure,… Expand

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On a smooth algebraic curve X with genus greater than 1 we consider a flat principal bundle with a reductive structure group S and a vector bundle associated with it. To this set of information we… Expand

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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 proper… Expand

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We show that if X is any proper complex variety, there is a weight decomposition on the real schematic homotopy type, in the form of an algebraic Gm-action. This extends to a mixed Hodge structure,… Expand

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Abstract We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed… Expand

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We define and construct mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic… Expand

#### References

SHOWING 1-10 OF 93 REFERENCES

The Hodge filtration on nonabelian cohomology

- Mathematics
- 1996

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 on… Expand

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 the… Expand

F-isocrystals and homotopy types

- Mathematics
- 2007

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 proper… Expand

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 strictly… Expand

Champs affines

- 2006

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 algebraic… Expand

Representative functions on discrete solvable groups

- Mathematics
- 1987

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 groups… Expand

Local Projective Model Structures on Simplicial Presheaves

- Mathematics
- 2001

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 category… Expand

The hodge de rham theory of relative malcev completion

- Mathematics
- 1996

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 associate… Expand

On the derived category of perverse sheaves

- Mathematics
- 1987

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

- Mathematics
- 1999

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 the… Expand