The author develops some aspects of derived analytic geometry as introduced by J. Lurie in [“Derived Algebraic Geometry V: Structured Spaces”, preprint, arXiv:0905.0459]. Therefore, a basic knowledge of Lurie’s work and the language of \(\infty\)-categories is required for reading the paper.
The paper starts by reviewing the notion of derived analytic space in the sense of Lurie’s work. Then, it is shown that there is a functor of points interpretation of the derived spaces obtained on a suitable site of derived Stein spaces. Next, the derived version of the analytication functor is introduced. This is a generalization of the functor studied by Serre extended to the category of derived stacks (more precisely derived Deligne-Mumford stack locally almost of finite presentation) over \(\mathbb{C}\), sending them to derived analytic stacks over \(\mathbb{C}\).
This derived analytification functor is used to prove the main results of the paper. These are a series of GAGA statements that compare algebraic and analytic notions on derived stacks. In particular, it is proved that for a proper morphism of analytic derived Artin stacks \(f: X \to Y\) the push-forward functor induces a functor between the \(\infty\)-categories of coherent sheaves \(f_*: \mathrm{Coh}^-(X) \to \mathrm{Coh}^-(Y)\), that for a morphism of algebraic derived Artin stacks \(f: X \to Y\) the identity \((-)^{\mathrm{an}} \circ f_* = f_* \circ (-)^{\mathrm{an}}\) holds, where \((-)^{\mathrm{an}}\) is the analytification functor, and that for an algebraic proper derived Artin stack \(X\) one has that \(\mathrm{Coh}(X) \cong\mathrm{Coh}(X^{\mathrm{an}})\).
These results are then applied for studying the essential image of the analytification functor and the deformation theory of derived analytic spaces. In particular, it is proved that a proper derived analytic space is in the image of \((-)^{\mathrm{an}}\) if and only if its classical underlying analytic space is, i.e. if and only if its \(0\)-th truncation is the analytification of an algebraic variety over \(\mathbb{C}\). Then, it proved that for any derived analytic space \(X\) and any \(x \in X\) the shifted tangent complex at \(x\), usually denoted \(\mathbb{T}_x X[-1]\), admits a differential graded Lie algebra structure.