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A multiple-volume book project: "Kuranishi spaces and Symplectic Geometry"

   I am writing a series of books titled "Kuranishi spaces and Symplectic Geometry", which develops the foundations for those areas of Symplectic Geometry which involve  "counting" moduli spaces of J-holomorphic curves -- Gromov-Witten invariants, Quantum Cohomology, Lagrangian Floer cohomology, Fukaya categories, Symplectic Field Theory, and so on. I hope that eventually there will be four or more volumes, which will go at least as far as full definitions of Gromov-Witten theory, Quantum Cohomology, Lagrangian Floer cohomology, and Fukaya categories. Currently only volumes I and II are partially written, which develop the theory of Kuranishi spaces, and you can download them from the links below. I will continue to post updates on this page as I write more. But the whole thing will take some years, so don't hold your breath.

'Kuranishi spaces and Symplectic Geometry. Volume I. Basic theory of (m-)Kuranishi spaces', pdf file.

'Kuranishi spaces and Symplectic Geometry. Volume II. Differential Geometry of (m-)Kuranishi spaces', pdf file.

'Kuranishi spaces and Symplectic Geometry. Volumes I and II together', pdf file.

Note: if you are reading the books on a computer, you may prefer the combined file of volumes I and II, as there are clickable links made using hyperref, including cross-references between volumes I and II.

  All comments on the project are welcome.

  Here is a longer description of the project:

  "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 are not well behaved -- so the theory of Kuranishi spaces was never developed very far.
   Around 2006 I became interested in the question of what is the "right" definition of Kuranishi space, which I felt Fukaya-Oh-Ohta-Ono had not found. In 2008 I understood the first part of the answer, when I read David Spivak's thesis on 'derived manifolds'. Scales fell from my eyes, and I immediately saw that:

Kuranishi spaces are really derived smooth orbifolds, where 'derived' is in the sense of the Derived Algebraic Geometry of Jacob Lurie and ToŽn-Vezzosi.

This has important consequences: we should define and study Kuranishi spaces using the ideas and methods of Derived Algebraic Geometry, including in particular higher categories, and sheaves / higher sheaves. As Fukaya-Oh-Ohta-Ono invented their Kuranishi spaces some years before Derived Algebraic Geometry began, they lacked essential tools.
  For my first version of a new definition of Kuranishi space, I took what was available on derived diffential geometry in the work of Lurie and Spivak and simplified it, to work in 2-categories rather than ∞-categories. The result was my theory of d-manifolds and d-orbifolds, which you can read about on another page on my website. This gives strict 2-categories dMan and dOrb of 'd-manifolds' and 'd-orbifolds', "derived "versions of smooth manifolds and smooth orbifolds, and also 2-categories
dManb, dOrbb of d-manifolds and d-orbifolds with boundary, and 2-categories dManc, dOrbc of d-manifolds and d-orbifolds with corners. All these are defined as examples of derived C-schemes and derived C-stacks, kinds of derived scheme and derived stacks over smooth functions. I worked on this theory over 2008-2012. I am revising a book on it, to be published by OUP, circa 2018.
  In 2014 I realized that there is an alternative way to define Kuranishi spaces, much closer to the original Fukaya-Oh-Ohta-Ono definition, equivalent to my 2-categories to my 2-categories dMan and dOrb of 'd-manifolds' and 'd-orbifolds'. To see why there should be two definitions, consider the following two equivalent definitions of manifolds:
  1. A manifold is a Hausdorff, second countable topological space X equipped with an atlas of pairwise compatible charts {(Vii): iI}.
  2. A manifold is a Hausdorff, second countable topological space X equipped with a sheaf OX of ℝ-algebras or C-rings locally isomorphic to (ℝn,On).
If we try to define derived manifolds by generalizing definition 1, we get Kuranishi spaces (or something similar). If we try to define them by generalizing definition 2, we get something like Spivak's derived manifolds, or my d-manifolds.
  I wrote up my new definition of Kuranishi spaces in arXiv:1409.6908, surveyed in arXiv:1510.07444. These define an ordinary category μKur of 'μ-Kuranishi spaces' which is equivalent to the homotopy category Ho(dMan) of the strict 2-category of d-manifolds dMan, and a weak 2-category mKur of 'm-Kuranishi spaces' which is equivalent to dMan, and a weak 2-category Kur of 'Kuranishi spaces', which is equivalent to the strict 2-category dOrb of d-orbifolds. They also defines boundary and corner versions μKurbμKurc, mKurb, mKurc, Kurb, Kurc.
  In the book series, I take the point of view that the definition of Kuranishi spaces takes as input a category Ḿan of 'manifolds' satisfying some axioms, and gives as output (2-)categories μḰur,mur,ur of '(μ-and m-)Kuranishi spaces'. Here Ḿan could be the usual category of classical manifolds Man, and then μḰur,mur,ur would be the usual (2-)categories μKur,mKur,Kur. But there are lots of other possibilities for Ḿan, including the category of manifolds with corners Manc and many variations of this, categories of manifolds with singularities, and so on. So we define many (2-)categories of (μ-and m-)Kuranishi spaces. In the C-algebraic geometry approach this would be much harder to do, as the category of classical manifolds Man is built into the definition of C-ring, at the very bottom level of the theory.

 I am currently planning four or more volumes in the series, as follows:

Volume I. Basic theory of (m-)Kuranishi spaces. Definitions of the category μḰur of μ-Kuranishi spaces, and the 2-categories mur of m-Kuranishi spaces and ur of Kuranishi spaces, over a category of 'manifolds' Ḿan such as classical manifolds Man or manifolds with corners Manc . Boundaries, corners, and corner (2-)functors for (μ-and m-)Kuranishi spaces with corners. Relation to similar structures in the literature, including Fukaya-Oh-Ohta-Ono's Kuranishi spaces, and Hofer-Wysocki-Zehnder's polyfolds. 'Kuranishi moduli problems', our approach to putting Kuranishi structures on moduli spaces, canonical up to equivalence.

Volume II. Differential Geometry of (m-)Kuranishi spaces. Tangent and obstruction spaces for (μ-and m-)Kuranishi spaces. Canonical bundles and orientations. (W-)tran sversality, (w-)submersions, and existence of w-transverse fibre products in mur and ur. M-(co)homology of manifolds and orbifolds as in my arXiv:1509.05672, virtual (co)chains and virtual (co)cycles for compact, oriented (m-)Kuranishi spaces in M-(co)homology. Orbifold strata of Kuranishi spaces. Bordism and cobordism for (m-)Kuranishi spaces.

Volume III. Kuranishi structures on moduli spaces of J-holomorphic curves. For very many moduli spaces of J-holomorphic curves M of interest in Symplectic Geometry, including singular curves, curves with Lagrangian boundary conditions, marked points, etc., we show that M can be made into a Kuranishi space M, uniquely up to equivalence in ur. We do this by a new method using 2-categories, similar to Grothendieck's 'representable functor' approach to moduli spaces in Algebraic Geometry. We do the same for many other classes of moduli problems for nonlinear elliptic p.d.e.s, including gauge theory moduli spaces. Natural relations between moduli spaces, such as maps Fi : Mk+1Mk forgetting a marked point, correspond to relations between the Kuranishi spaces, such as a 1-morphism Fi : Mk+1Mk in ur. We discuss orientations on Kuranishi moduli spaces.

Volumes IV-. Big theories in Symplectic Geometry. To include Gromov-Witten invariants, Quantum Cohomology, Lagrangian Floer cohomology, and Fukaya categories.

References

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', to appear in Memoirs of the A.M.S.
-- Also available on the Web as arXiv:1001.0023.

D. Joyce, 'An introduction to C-schemes and C-algebraic geometry', pages 299-325 in H.-D. Cao and S.-T. Yau, editors, Surveys in Differential Geometry 17 (2012), Lectures given at the JDG symposium, June 2010, in memory of C.C. Hsiung.
-- Also available on the Web as arXiv:1104.4951.

D. Joyce, 'An introduction to d-manifolds and derived differential geometry', pages 230-281 in L. Brambila-Paz, O. Garcia-Prada, P. Newstead and R.P. Thomas, editors, 'Moduli spaces', London Mathematical Society Lecture Note Series 411, Cambridge University Press, 2014.
-- Also available on the Web as arXiv:1206.4207.

D. Joyce, 'D-manifolds, d-orbifolds and derived differential geometry: a detailed summary', arXiv:1208.4948, 2012. 

D. Joyce, 'D-manifolds and d-orbifolds: a theory of derived differential geometry', preliminary version of book (2012) pdf file. 768 pages.

D. Joyce, 'A new definition of Kuranishi space', arXiv:1409.6908, 2014.

D. Joyce, 'Some new homology and cohomology theories of manifolds and orbifolds', arXiv:1509.05672, 2015.

D. Joyce, 'Kuranishi spaces as a 2-category', arXiv:1510.07444, 2015.

D.I. Spivak, 'Derived smooth manifolds', Duke Mathematical Journal 153 (2010), 55-128. arXiv:0810.5174.