Schematic homotopy types and non-abelian Hodge theory

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.