Schematic homotopy types and non-abelian Hodge theory

Abstract

In this work we use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the \textit{schematization functor} $X\mapsto (X\otimes \mathbb{C}{)}^{sch}$, 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 \textit{Hodge decomposition} on $(X\otimes \mathbb{C}{)}^{sch}$. This Hodge decomposition is encoded in an action of the discrete group ${\mathbb{C}}^{\times \delta }$ on the object $(X\otimes \mathbb{C}{)}^{sch}$ and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group as defined by C.Simpson, and in the simply connected case, the Hodge decomposition on the complexified homotopy groups as defined by J.Morgan and R. Hain. This Hodge decomposition is shown to satisfy 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 a first application we construct a new family of examples of homotopy types which are not realizable as complex projective manifolds. Our second application is a formality theorem for the schematization of a complex projective manifold. Finally, we present conditions on a complex projective manifold $X$ under which the image of the Hurewitz morphism of ${\pi }_i(X)\to H_i(X)$ is a sub-Hodge structure.

57 pages. This new version has been globally reorganized and includes additional results and applications. Minor corrections

Cited by in corpus (9)

• Higher and derived stacks: a global overview
• Galois actions on homotopy groups
• Algebraic and topological aspects of the schematization functor
• Non-abelian real Hodge theory for proper varieties
• Tannaka duality over ring spectra
• Formality and splitting of real non-abelian mixed Hodge structures
• The Hodge theoretic fundamental group and its cohomology
• The de Rham homotopy theory and differential graded category
• Rational homotopy theory and differential graded category