Moduli of connections on smooth varieties

Overview & Synopsis

This paper is a companion to [Pa-To]. 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 ∞ of the Betti topological space associated to X studied in [Pa-To]. We explain how to construct two derived moduli functors Vect (X) and Vect (∂X), of flat bundles on X and on ∂X, as well as a restriction map R : Vect (X) −→ Vect (∂X). This work contains two main results. First of all 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 studied in [Pa-To]. 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.