# 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

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