Moduli of objects in dg-categories

To any dg-category $T$ (over some base ring $k$), we define a $D^{-}$-stack $\mathcal{M}_{T}$ in the sense of \cite{hagII}, classifying certain $T^{op}$-dg-modules. When $T$ is saturated, $\mathcal{M}_{T}$ classifies compact objects in the triangulated category $[T]$ associated to $T$. The main result of this work states that under certain finiteness conditions on $T$ (e.g. if it is saturated) the $D^{-}$-stack $\mathcal{M}_{T}$ is locally geometric (i.e. union of open and geometric sub-stacks). As a consequence we prove the algebraicity of the group of auto-equivalences of a saturated dg-category. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as well as complexes of representations of a finite quiver.Comment: 64 pages. Minor corrections. Section 3.4 including some corollaries has been added. Sections 1 and 2.5 added, as well as some remarks. To appear in Annales de l'EN

Toward a Galoisian interpretation of homotopy theory

Given any pointed CW complex (X,x), it is well known that the fondamental group of X pointed at x is naturally isomorphic to the automorphism group of the functor which associates to a locally constant sheaf on X its fibre at x. The purpose of this work is to generalize this fact to higher homotopy. For this we introduce the (infinite) category of locally constant stacks on X, and we prove that the loop-space of endomorphisms of its fibre functor at x is naturally equivalent to the loop space of X based at x.Comment: French, 38 pages. To appear in "Cahiers de topologie et geometrie differentielle categoriques

Schematic homotopy types and non-abelian Hodge theory

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