Schematic homotopy types and non-abelian Hodge theory

L. Katzarkov, T. Pantev, B. Toën

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

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 2008

Keywords

  • Homotopy type
  • Non-abelian Hodge theory

ASJC Scopus subject areas

  • Algebra and Number Theory

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