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Algebraic and topological aspects of the schematization functor

Part of: Compact analytic spaces Homotopy theory Cycles and subschemes

Published online by Cambridge University Press:  01 May 2009

L. Katzarkov ,
T. Pantev  and
B. Toën
L. Katzarkov
Affiliation:
Department of Mathematics, University of Miami, 1365 Memorial Drive, Ungar 515, Coral Gables, FL 33146, USA (email: [email protected])
T. Pantev
Affiliation:
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA (email: [email protected])
B. Toën
Affiliation:
Laboratoire Emile Picard, Université Paul Sabatier Bat 1R2, 31062 Toulouse Cedex 9, France (email: [email protected])
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Abstract

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We study some basic properties of schematic homotopy types and the schematization functor. We describe two different algebraic models for schematic homotopy types, namely cosimplicial Hopf alegbras and equivariant cosimplicial algebras, and provide explicit constructions of the schematization functor for each of these models. We also investigate some standard properties of the schematization functor that are helpful for describing the schematization of smooth projective complex varieties. In a companion paper, these results are used in the construction of a non-abelian Hodge structure on the schematic homotopy type of a smooth projective variety.

Keywords

schematic homotopy typeshomotopy theorynon-abelian Hodge theory

MSC classification

Secondary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 32J27: Compact Kähler manifolds 55P62: Rational homotopy theory
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 633 - 686
Copyright
Copyright © Foundation Compositio Mathematica 2009

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