I am working on a new book. If you are interested, the best place to start is with the following survey paper:
D. Joyce, 'An introduction to d-manifolds and derived differential geometry', arXiv:1206.4207, version 3, December 2012. 45 pages.
Here too is a "detailed summary" of the book, a longer survey paper, which basically corresponds to Chapter 1 and Appendix A of the current version of the book, and contains more-or-less all the main definitions and theorems, but no proofs.
D. Joyce, 'D-manifolds, d-orbifolds and derived differential geometry: a detailed summary', arXiv:1208.4948, version 2, December 2012. 173 pages.
You can download a preliminary version of the book here:
'D-manifolds and d-orbifolds: a theory of derived differential geometry', pdf file. 768 pages.
You can also download the slides of a talk I gave on it in Luminy in July 2012 here:
'D-manifolds and derived differential geometry'. pdf file
and the slides of three talks on 'D-manifolds and d-orbifolds: a theory of derived differential geometry', which I gave in Tokyo in July 2012:
first talk pdf file
second talk pdf file
third talk pdf file
Here is a survey paper on C∞-schemes and C∞-algebraic geometry which may be useful: arXiv:1104.4951.
When it's finished I'll try and persuade someone to publish it. This is work in progress. Please do send me comments about it, mistakes, etc.
Here is a brief description of what the book is about. See the reference list at the end for relevant literature.
"Kuranishi spaces" are a class of geometric spaces introduced in 1990 by Fukaya and Ono, as the geometric structure on moduli spaces of J-holomorphic curves in a symplectic manifold, and used in the work of Fukaya, Oh, Ohta and Ono on Lagrangian Floer cohomology and Fukaya categories. Although their definition was sufficient for their applications, it did not give a very satisfactory notion of geometric space -- notions of morphisms, or even of when two Kuranishi spaces are "the same", are not well behaved -- so the theory of Kuranishi spaces was never developed very far. The book began as a project to find the "right" definition of Kuranishi space, which I believe I have done.
The book describes and studies a new class of geometric objects I call "d-manifolds". D-manifolds are a kind of "derived" smooth manifold, where "derived" is in the sense of the derived algebraic geometry of Jacob Lurie, Bertrand Toen, etc. The closest thing to them in the literature is the "derived manifolds" of David Spivak. But d-manifolds are rather simpler than Spivak's derived manifolds -- d-manifolds form a 2-category which is constructed using fairly basic techniques from algebraic geometry, but derived manifolds form an ∞-category (simplicial category) which uses advanced ideas like homotopy sheaves and Bousfeld localization.
Following Spivak, my d-manifolds are based on the idea of "C∞-schemes". These were invented in Synthetic Differential Geometry (see Dubuc, 1981, and my 2010 paper below), and are a way of applying the methods of Algebraic Geometry in the world of real smooth geometry to get a large class of geometric objects which include smooth manifolds, but also many singular spaces.
Manifolds are examples of d-manifolds -- that is, the category of manifolds embeds as a subcategory of the 2-category of d-manifolds -- but d-manifolds also include many spaces one would regard classically as singular or obstructed. A d-manifold has a virtual dimension, an integer, which may be negative. Almost all the main ideas of differential geometry have analogues for d-manifolds -- submersions, immersions, embeddings, submanifolds, orientations, transverse fibre products, and so on -- but the derived versions are often stronger. For example, the intersection of two submanifolds in a manifold exists as a manifold if the intersection is transverse, but it always exists as a d-manifold. There are also good notions of d-manifolds with boundary and d-manifolds with corners, and orbifold versions of all this, d-orbifolds. I claim that the (morally and aesthetically) "right" definition of Kuranishi space in the work of Fukaya-Oh-Ohta-Ono is that they are d-orbifolds with corners.
A useful property of d-manifolds and d-orbifolds is that they have well-behaved virtual cycles or virtual chains. So, for example, if X is a compact oriented d-manifold of virtual dimension k, and Y is a manifold, and f : X →Y is a 1-morphism, then we can define a virtual class [X] in the homology group Hk(Y; Z), which is unchanged under deformations of X,f. This will be important in applications of d-manifolds and d-orbifolds.
Many important areas of mathematics involve "counting" moduli spaces of geometric objects to define enumerative invariants or homology theories -- for instance, Donaldson and Seiberg-Witten invariants for 4-manifolds, Donaldson-Thomas invariants of Calabi-Yau 3-folds, Gromov-Witten invariants in algebraic or symplectic geometry, instanton Floer cohomology, Lagrangian Floer cohomology, contact homology, Symplectic Field Theory, Fukaya categories, ... In all of these (at least over C for the algebraic geometry cases, and away from reducible connections, etc.) the moduli spaces concerned will be oriented d-manifolds or d-orbifolds, and the counting can be done using virtual cycles or chains. So, d-manifolds and d-orbifolds provide a unified way of looking at these counting problems. There are truncation functors from geometric structures currently used to define virtual classes to d-manifolds and d-orbifolds. For instance, any moduli space of solutions of a smooth nonlinear elliptic p.d.e. on a compact manifold is a d-manifold. In algebraic geometry, a C-scheme with a perfect obstruction theory can be made into a d-manifold. In symplectic geometry, Kuranishi spaces and polyfold structures on moduli spaces of J-holomorphic curves induce d-orbifold structures.
D. Borisov, 'Derived manifolds and d-manifolds', arXiv:1212.1153.
E.J. Dubuc, 'C∞-schemes', Amer. J. Math. 103 (1981), 683-690.
K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, 'Lagrangian intersection Floer theory - anomaly and obstruction', Parts I and II. AMS/IP Studies in Advanced Mathematics, 46.1 and 46.2, A.M.S./International Press, 2009.
K. Fukaya and K. Ono, 'Arnold Conjecture and Gromov-Witten invariant', Topology 38 (1999), 933-1048.
D. Joyce, 'Algebraic Geometry over C∞-rings', arXiv:1001.0023, 2010.
J. Lurie, 'Derived Algebraic Geometry I: Stable ∞-categories', math.CT/0608228, 2006.
J. Lurie, 'Derived Algebraic Geometry V: Structured spaces', arXiv:0905.0459, 2009.
D.I. Spivak, 'Derived smooth manifolds', Duke Mathematical Journal 153 (2010), 55-128. arXiv:0810.5174.