We define a new geometric object--the stack of local systems with restricted variation. We formulate a version of the categorical geometric Langlands conjecture that makes sense for any constructible sheaf theory (such as l-adic sheaves). We formulate a conjecture that makes precise the connection between the category of automorphic sheaves and the space of automorphic functions.

The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support

The aim of this paper is to study the pro-algebraic fundamental group of a compact Kaehler manifold. Following work by Simpson, the structure of this group's pro-reductive quotient is already well understood. We show that Hodge-theoretic methods can also be used to establish that the pro-unipotent radical is quadratically presented. This generalises both Deligne et al.'s result on the de Rham fundamental group, and Goldman and Millson's result on deforming representations of Kaehler groups, and can be regarded as a consequence of formality of the schematic homotopy type. New examples are given of groups which cannot arise as Kaehler groups.

The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kaehler manifold

We develop Tannaka duality theory for dg categories. To any dg functor from a dg category A to finite-dimensional complexes, we associate a dg coalgebra C via a Hochschild homology construction. When the functor is faithful, this gives a quasi-equivalence between the derived dg categories of A-modules and of C-comodules. When A is Morita fibrant (i.e. an idempotent-complete enhanced triangulated category), it is thus quasi-equivalent to the derived dg category of compact C-comodules. We give several applications for pro-algebraic homotopy types associated to various cohomology theories, and for motivic Galois groups.

Tannaka duality for enhanced triangulated categories

The goal of this paper is to develop the theory of Deligne-Beilinson cohomology of affine groups with a mixed Hodge structure. The motivation comes from Hodge theory and the study of motives, where such groups appear. Several of Francis Brown's period computations (arXiv:1407.5167) are interpreted as elements of the DB cohomology of the relative unipotent completion of $SL_2(Z)$ and their cup products. The results in this paper are used in arXiv:1403.6443 where they are used to prove that Pollack's quadratic relations are motivic.

Deligne-Beilinson Cohomology of Affine Groups

In the proof of the Freudenthal–Weil theorem in, for example (5), essential use is made of the fact that if G and H are compact analytic groups and o: G → H is a continuous epimorphism then o( Z(G) 0 ) = Z(H) 0 where the subscript 0 denotes the identity component of a topological group G and Z(G) its centre. Although this is sufficient for the proof of the Freudenthal–Weil theorem it raises the interesting question as to whether actually o( Z(G) ) = Z(H) (from which the above would follow) and, if so, in what generality this can be expected. The present paper deals with this question, in more general form, as well as certain of its structural consequences.

On pro-reductive groups

Pro-Affine Algebraic Groups

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Pro-Affine Algebraic Groups