2019/05/28 by Tony Pantev, Pantev, Tony, Bertrand Toën +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1905.12124
openalex publication_date 2019/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∇(X) and Vect∇(∂ X), of flat bundles on X and on ∂ X, as well as a restriction map R : Vect∇(X) → Vect^∇(∂ 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.