An introduction to d-manifolds and derived differential geometry (Q2875832)

The article under review serves as a survey to a book in preparation (by the same author) on d-manifolds and d-orbifolds, available on the author's webpage. It discusses a possible notion of \textit{derived smooth manifolds} to be used in a derived differential geometry. Such a derived differential geometry is an analytic counterpart to derived algebraic geometry, in the sense of Lurie or Toën-Vezzosi. The goal of this theory is to put the definition of a Kuranishi space [\textit{K. Fukaya} and \textit{K. Ono}, Topology 38, No. 5, 933--1048 (1999; Zbl 0946.53047)] on a firmer footing and apply it to study moduli spaces in symplectic geometry. Besides this short survey there is a longer 173-page survey, also available on the author's webpage.NEWLINENEWLINEThe setup of the problem is as follows: Let \((M,\omega)\) be a symplectic manifold, let \(J\) be a \(\omega\)-tame almost complex structure and \(\beta\) a class in \(\mathrm{H}_2(M,\mathbb{Z})\). One wishes to count (isomorphism classes of) Riemann surfaces of genus \(g\) with \(m\) marked points (and possibly mild singularities) such that the marked curve is stable (i.e.\ has finite automorphism group) and the class of the curve in homology is the fixed \(\beta\). This set can be given the structure of a manifold \(\mathcal{M}_{g,m}(J,\beta)\), and the goal is to understand this manifold when it has dimension zero, hence to count the number of curves as a solution to an enumerative problem. Less ideally, one wishes to ``count'' curves when this manifold only has \textit{expected} dimension zero while the actual dimension is higher. To do this one needs to equip \(\mathcal{M}_{g,m}(J,\beta)\) with extra structure and study it using virtual cycles.NEWLINENEWLINEThis phenomenon has analogues in (derived) algebraic geometry, where the extra structure takes the form of an obstruction theory, or related constructions such as the cotangent complex. The philosophy of ``Kontsevich's hidden smoothness'' is that a singular moduli space is actually a shadow or truncation of a better behaved derived moduli space, that is more appropriate to study. In the symplectic world however there are competing notions of which extra structure one uses. The ``minimal'' such approach using the least amount of data are Kuranishi spaces, i.e.\ one wishes to equip \(\mathcal{M}_{g,m}(J,\beta)\) with the structure of a Kuranishi space. However, the theoretical framework for these spaces lacks some foundational work. The author explains how his notion of \textit{d-orbifolds with corners} provides the foundations in which one can study \(\mathcal{M}_{g,m}(J,\beta)\) as Kuranishi spaces.NEWLINENEWLINEHis notion of d-manifolds (and related notions including orbifold structure, or boundaries and corners) is a \textit{2-categorical truncation} of Spivak's derived smooth manifolds [\textit{D. I. Spivak}, Duke Math. J. 153, No. 1, 55--128 (2010; Zbl 1420.57073)]. These derived smooth manifolds are a \(\infty\)-categorical notion, and their definition requires many technical preliminaries provided by Lurie's work on higher topos theory, higher algebra and derived algebraic geometry. Kuranishi spaces however are intended to constitute a minimal amount of extra data. Therefore the author develops a framework of d-manifolds in the more accessible language of 2-categories, and explains how this already provides enough data to study Kuranishi spaces and the enumerative problems related to \(\mathcal{M}_{g,m}(J,\beta)\).NEWLINENEWLINEThe article is organised as follows. After an introduction, Section~5.2 discusses the notion of \(C^\infty\)-rings, which are commutative rings equipped with extra structure that describes their compatibility with smooth functions \(f:\mathbb{R}^n\to\mathbb{R}\). These are the building blocks for \(C^\infty\)-schemes as particular \(C^\infty\)-ringed spaces, much like commutative rings are the building blocks for usual schemes as ringed spaces. The main example of such a \(C^\infty\)-ring is the ring \(C^\infty(X)\) of smooth functions \(X\to\mathbb{R}\) on a manifold, together with its canonical structure as a \(C^\infty\)-ring. Hence many interesting examples of \(C^\infty\)-schemes turn out to be affine in this sense, unlike in algebraic geometry where affine schemes certainly do not cover most of the interesting cases. The notions of modules over \(C^\infty\)-rings, with cotangent modules as an important example, and their global counterparts are also discussed. These serve as the underived aspect of the theory that is developed in later sections, and constitute a formalisation of differential geometry in a language used by algebraic geometers.NEWLINENEWLINEIn Section~5.3 the category \(\pmb{\mathrm{dSpa}}\) of d-spaces is introduced. This is a 2-category that constitutes a derived version of \(C^\infty\)-schemes. The 2-categorical notions that make up the theory are discussed, the main ingredient being that one replaces the cotangent sheaf by a \textit{virtual cotangent sheaf}, which boils down to a morphism of sheaves. This is a manifestation of the \textit{cotangent complex}, which in other incarnations of derived algebraic geometry such as Lurie's or Toën-Vezzosi's can live in many degrees, whereas for the quasi-smooth objects that should constitute a derived differential geometry it is concentrated in 2~degrees, hence is nothing but a morphism. The technicalities of ``gluing up to homotopy'' that make derived algebraic geometry a sometimes difficult area are thereby greatly reduced, circumventing the need for \(\infty\)-category theory. It is shown that all fibre products exists \(\pmb{\mathrm{dSpa}}\) using the explicit approach to gluings in this 2-category.NEWLINENEWLINEThe main aspects of the theory of d-manifolds and derived differential geometry are contained in Section~5.4. Here the category \(\pmb{\mathrm{dMan}}\) of d-manifolds is introduced as a subcategory of \(\pmb{\mathrm{dSpa}}\) consisting of derived spaces that locally look like fiber products of usual manifolds (taken in the category of d-spaces). This way the notion of d-manifolds is explicitly set up to make fibre products of manifolds (even if they are non-transverse!) work. This is the essential ingredient of the theory, which is parallel to the fact in derived algebraic geometry that the inclusion functor of schemes into derived schemes does not commute with fibre products.NEWLINENEWLINEThe remainder of the section consists of introducing derived analogues of well-known notions in classical differential geometry: virtual quasicoherent sheaves, virtual vector bundles, étale morphisms, \dots\ It turns out that there are a weak and a strong version of submersions, immersions and embeddings, but for a morphism induced from a smooth map of manifolds the weak and strong version agree. Fibre products of d-manifolds and a notion of d-transversality are discussed, the main application that the author has in mind being fibre products of d-manifolds over a usual manifold. An analogue of Whitney's embedding theorem for d-manifolds, and a generalisation of an orientation of a manifold are discussed.NEWLINENEWLINEThen the author goes on to introduce the derived analogues of manifolds with boundaries and corners, and orbifolds with boundaries and corners. Not much of the technicalities involved in their definition are discussed. Instead the author focusses on an important application of d-manifolds: to obtain a better theory of bordism and virtual cycles, avoiding the issues with transversality that one encounters in the underived setting. The virtual classes that one obtains in this theory are constructions to help in tackling enumerative problems, the important fact being that they show up \textit{naturally} in the theory. The section concludes with discussions of the relationships between d-manifolds and other concepts in the literature, and gives many examples of appropriate d-manifold structures on moduli spaces studied in the literature. The conclusion is that d-manifolds are a \textit{concrete} approach to derived differential geometry that in some sense is \textit{minimal}.NEWLINENEWLINEIn an appendix a short introduction to the language of 2-categories is given.NEWLINENEWLINEFor the entire collection see [Zbl 1286.14002].