### 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 language | English (US) |
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Pages (from-to) | 582-632 |

Number of pages | 51 |

Journal | Compositio Mathematica |

Volume | 144 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2008 |

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### Keywords

- Homotopy type
- Non-abelian Hodge theory

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Compositio Mathematica*,

*144*(3), 582-632. https://doi.org/10.1112/S0010437X07003351