Compositio Mathematica

Research Article

Schematic homotopy types and non-abelian Hodge theory

L. Katzarkova1, T. Panteva2 and B. Toëna3

a1 Department of Mathematics, University of Miami, College of Arts and Sciences, PO Box 249085, Coral Gables, FL 33124-4250, USA (email: l.katzarkov@math.miami.edu)

a2 Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA (email: tpantev@math.upenn.edu)

a3 Laboratoire Emile Picard, Université Paul Sabatier, Bat 1R2, 31062 Toulouse Cedex 9, France (email: toen@math.univ-toulouse.fr)

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 decomposition is encoded in an action of the discrete group $\mathbb {C}^{\times \delta }$ on the object $(X\otimes \mathbb {C})^{\mathrm {sch}}$ 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.

(Received October 24 2006)

(Revised July 03 2007)

(Accepted August 24 2007)

2000 Mathematics subject classification

  • 14C30;
  • 32J27;
  • 55P62

Keywords

  • non-abelian Hodge theory;
  • homotopy type

Footnotes

The first author was partially supported by NSF Career Award DMS-9875383 and an A. P. Sloan research fellowship. The second author was Partially supported by NSF Grants DMS-0099715 and DMS-0403884 and an A. P. Sloan research fellowship.