Schematic homotopy types and non-abelian Hodge theory

L. Katzarkov, T. Pantev, B. Toën

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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 → (X ⊗ ℂ)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 ⊗ ℂ) sch. This Hodge decomposition is encoded in an action of the discrete group ℂ×δ on the object (X ⊗ ℂ) sch and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold.

Original languageEnglish (US)
Pages (from-to)582-632
Number of pages51
JournalCompositio Mathematica
Volume144
Issue number3
DOIs
StatePublished - May 1 2008

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Keywords

  • Homotopy type
  • Non-abelian Hodge theory

ASJC Scopus subject areas

  • Algebra and Number Theory

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