GAGA theorems in derived complex geometry

In this paper, we expand the foundations of derived complex analytic geometry introduced by Jacob Lurie in 2011. 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 X , the canonical map X a n → X X^{\mathrm {an}} \to 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 of the author and Tony Yue Yu, 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.