%0 Generic %1 pantev2019moduli %A Pantev, Tony %A Toën, Bertrand %D 2019 %K connections %T Moduli of flat connections on smooth varieties %U http://arxiv.org/abs/1905.12124 %X We study the moduli functor of flat bundles on smooth, possibly non-proper, algebraic variety $X$ (over a field of characteristic zero). For this we introduce the notion of formal boundary of $X$, denoted by $X$, which is a formal analogue of the boundary at infinity of the Betti topological space associated to $X$. We explain how to construct two derived moduli functors $Vect^\nabla(X)$ and $Vect^\nabla(X)$, of flat bundles on $X$ and on $X$, as well as a restriction map $R : Vect^\nabla(X) Vect^\nabla(X)$ from the former to the later. This work contains two main results. First we prove that the morphism R comes equipped with a canonical shifted Lagrangian structure in the sense of PTVV. This first result can be understood as the de Rham analogue of the existence of Poisson structures on moduli of local systems previously studied by the authors. As a second statement, we prove that the geometric fibers of $R$ are representable by "quasi-algebraic spaces", a slight weakening of the notion of algebraic spaces.

@misc{pantev2019moduli, abstract = {We study the moduli functor of flat bundles on smooth, possibly non-proper, algebraic variety $X$ (over a field of characteristic zero). For this we introduce the notion of \emph{formal boundary} of $X$, denoted by $\partial X$, which is a formal analogue of the boundary at infinity of the Betti topological space associated to $X$. We explain how to construct two derived moduli functors $Vect^{\nabla}(X)$ and $Vect^{\nabla}(\partial X)$, of flat bundles on $X$ and on $\partial X$, as well as a restriction map $R : Vect^{\nabla}(X) \rightarrow Vect^\nabla(\partial X)$ from the former to the later. This work contains two main results. First we prove that the morphism R comes equipped with a canonical shifted Lagrangian structure in the sense of [PTVV]. This first result can be understood as the de Rham analogue of the existence of Poisson structures on moduli of local systems previously studied by the authors. As a second statement, we prove that the geometric fibers of $R$ are representable by "quasi-algebraic spaces", a slight weakening of the notion of algebraic spaces.}, added-at = {2019-05-30T22:25:45.000+0200}, author = {Pantev, Tony and Toën, Bertrand}, biburl = {https://www.bibsonomy.org/bibtex/2e4fc962197beec8c27198385383480f2/simonechiarello}, description = {Moduli of flat connections on smooth varieties}, interhash = {f1a8887162e0dcd14999aa6d66649c2a}, intrahash = {e4fc962197beec8c27198385383480f2}, keywords = {connections}, note = {cite arxiv:1905.12124Comment: 53 pages. First draft}, timestamp = {2019-05-30T22:25:45.000+0200}, title = {Moduli of flat connections on smooth varieties}, url = {http://arxiv.org/abs/1905.12124}, year = 2019 }