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We study some basic properties of schematic homotopy types and the schematization functor. We describe two different algebraic models for schematic homotopy types, namely cosimplicial Hopf alegbras and equivariant cosimplicial algebras, and provide explicit constructions of the schematization functor for each of these models. We also investigate some standard properties of the schematization functor that are helpful for describing the schematization of smooth projective complex varieties. In a companion paper, these results are used in the construction of a non-abelian Hodge structure on the schematic homotopy type of a smooth projective variety.
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, 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 . This Hodge decomposition is encoded in an action of the discrete group on the object 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.
For a long time it was not known if the groups which appear as fundamental groups of projective manifolds are similar to linear groups. First J. P. Serre asked if there are smooth complex projective manifolds with nonresidually finite fundamental groups. D. Toledo  found examples of nonresidually finite groups which appear as fundamental groups of complex projective manifolds. Further examples were given by Catanese, Kollàr and Nori . We shall call a group Kähler if it can be realized as a fundamental group of a compact Kähler manifold. We assume (this is a popular belief) that Kähler groups can be realized as fundamental groups of projective manifolds and hence by the Lefschetz hyperplane section theorem they can also be realized as fundamental groups of complex projective surfaces.
The problem, which arises naturally is to find some “natural borders” for the class of groups which appear as fundamental groups of projective (Kähler manifolds) inside the class of finitely presented groups. In this paper we are going to present a construction which shows that fundamental groups of projective surfaces (and hence Kähler groups) are rather densely distributed among all finitely presented groups.
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