Higher analytic stacks and GAGA theorems

Abstract

We develop the foundations of higher geometric stacks in complex analytic geometry and in non-archimedean analytic geometry. We study coherent sheaves and prove the analog of Grauert's theorem for derived direct images under proper morphisms. We define analytification functors and prove the analog of Serre's GAGA theorems for higher stacks. We use the language of infinity category to simplify the theory. In particular, it enables us to circumvent the functoriality problem of the lisse-étale sites for sheaves on stacks. Our constructions and theorems cover the classical 1-stacks as a special case.for the relative case in complex geometry would be more involved because Stein algebras are not noetherian in general. Related works.In the classical sense, complex analytic stacks were considered in [4], and non-archimedean analytic stacks were considered in [49,47] to the best of our knowledge.The general theory of higher stacks was studied extensively by Simpson [42], Lurie [33] and Toën-Vezzosi [45,46]. Our Section 2 follows mainly [46]. However, we do not borrow directly the HAG context of [46], because the latter is based on symmetric monoidal model categories which is not suitable for analytic geometry.Our definition of properness for analytic stacks follows an idea of Kiehl in rigid analytic geometry [26]. The coherence of derived direct images under proper morphisms (i.e. Grauert's theorem) was proved in [17,27,15,23,30] for complex analytic spaces and in [26] for rigid analytic spaces.Analytification of algebraic spaces and classical algebraic stacks was studied in [1,31,44,10]. Analogs and generalizations of Serre's GAGA theorems are found in [19,28,37,5,9,8,21]. Our proofs use induction on the geometric level of higher stacks. We are very much inspired by the strategies of Brian Conrad in [8].In [39], Mauro Porta deduced from this paper a GAGA theorem for derived complex analytic stacks. Several results in this paper are also used in the work [40].In [50], Tony Yue Yu applied the GAGA theorem for non-archimedean analytic stacks to the enumerative geometry of log Calabi-Yau surfaces.Acknowledgements. We are grateful to for very useful discussions. The authors would also like to thank each other for the joint effort.2. Higher geometric stacks 2.1. Notations. We refer to Lurie [32,36] for the theory of ∞-category. The symbol S denotes the ∞-category of spaces. Definition 2.1 (cf. [43, 00VH]). An ∞-site (C, τ ) consists of a small ∞-category C and a set τ of families of morphisms with fixed target {U i → U } i∈I , called τ -coverings of C, satisfying the following axioms:Remark 2.3. In all the examples that we will consider in this paper, the site (C, τ ) is a classical Grothendieck site, i.e. the category C is a 1-category. We state Definition 2.2 for general ∞-sites because it will serve as a geometric context for derived stacks in our subsequent works (cf. [39,40]). Proposition 2.4. Let (C, τ ) be an ∞-site and let D be an (n + 1, 1)-category for n ≥ 0 (cf. [32, §2.3.4]). Then a functor F : C op → D satisfies d...

Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I

“…We use non-archimedean geometry here to cut out relevant components of the moduli space of stable maps. Finally, thanks to the GAGA theorem for non-archimedean analytic stacks proved in [45], we are able to resort to the virtual fundamental classes in algebraic geometry and achieve the enumeration.…”

“…Our previous works on non-archimedean geometry and tropical geometry [51,49,50,52,45] provide general foundations for the context of this paper. We will refer to [49, §6] and [45] for the theory of stacks in non-archimedean analytic geometry.…”

Dagger geometry as Banach algebraic geometry

“…An advantage of this foundational approach will be towards a theory of analytic stacks related to recent work ( [12], [29], [36]) , analytic D-modules (as in [3]), constructible sheaves, Arakelov Theory and more. In future work, we will look at geometry over the Adeles in this context and develop a form of descent which can prove results for schemes over the analytic integers by proving them over R and the Q p .…”

A differential graded model for derived analytic geometry

“…By taking the opposite model category DG + Aff T (or a pseudo-model subcategory) to nonnegatively graded T-DGAs as our analogue of affine schemes, we can now develop the theory of derived stacks (by analogy with [TV,Lur1]) with respect to any precanonical Grothendieck topology, and a suitable class C of morphisms, for any rational Fermat theory T. In the algebraic setting, the most common classes of morphisms to consider areétale surjections (giving rise to Deligne-Mumford stacks) and smooth surjections (giving rise to Artin stacks). The general properties such a class of morphisms must satisfy are summarised in [Pri1,Properties 1.8] or [PY1,Definition 2.10], giving rise to the theory of n-geometric stacks as certain simplicial presheaves on non-negatively graded T-DGAs (beware of discrepancies in the value of n between references, stemming from different versions of [TV]).…”