GAGA theorems in derived complex geometry

Abstract

In this paper, we expand the foundations of derived complex analytic geometry introduced in [10]. We start by studying the analytification functor and its properties. In particular, we prove that for a derived complex scheme locally almost of finite presentation X, the canonical map Xan → X is flat in the derived sense. Next, we provide a comparison result relating derived complex analytic spaces to geometric stacks. Using these results and building on the previous work [20], we prove a derived version of the GAGA theorems. As an application, we prove that the infinitesimal deformation theory of a derived complex analytic moduli problem is governed by a differential graded Lie algebra.