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Kuranishi Structures and Virtual Fundamental Chains

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Abstract

The technique of virtual fundamental cycles (and chains) was introduced in the year 1996 by several groups of mathematicians [FOn2, LiTi2, LiuTi, Ru2, Sie] to provide a differential geometric foundation on the study of moduli spaces of pseudo-holomorphic curves entering on the one hand in the Gromov–Witten theory and on the other in the study of the Arnold conjecture in symplectic geometry without the assumption of any kind of positivity on general compact symplectic manifolds.

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Notes

  1. 1.

    In our main application, such a system is constructed from the moduli spaces of pseudo-holomorphic curves. However, those results we mention here are independent of the origin of such a system.

  2. 2.

    It is characterized by \(i(X_{H_t})\omega = d{H_t}\).

  3. 3.

    See for example [FOn1, Section 3].

  4. 4.

    This is mentioned also by Joyce [Jo1].

  5. 5.

    Except the choice of smooth structure of gluing parameter.

  6. 6.

    This is the viewpoint taken and insisted upon by Grothendieck.

  7. 7.

    The theory of singularities of C ∞ functions is one typical example.

  8. 8.

    For this reason in many places we do not need to say much about the proof.

  9. 9.

    Thorough knowledge of such a technicality, of course, should be shared among the people whose interest lies also on extending the technology to the extreme of its potential border and/or using the most delicate and difficult case of the technology to obtain the sharpest results possible.

  10. 10.

    Introduction to Part II can be seen at the beginning of Part II.

  11. 11.

    The authors thank D. Joyce for drawing our attention to this point.

  12. 12.

    Such a figure was observed in the year 1996 when the idea of virtual fundamental chains and cycles was incepted. (See [FOn2, Fig. 4.8].)

  13. 13.

    Here W is a vector space, ω is a top degree compactly supported form on W and \(\mathfrak s^{ \epsilon }\) is a section of the finite-dimensional bundle on \(W \times U_{\mathfrak p}\), which is the pullback of \(\mathcal E_{\mathfrak p}\).

  14. 14.

    Statements on other variants of transversality also hold.

  15. 15.

    For the construction of Gromov–Witten invariants there is an approach by algebraic geometry, which is somewhat similar to the symplectic geometry approach. We do not discuss it here.

  16. 16.

    See Theorem 8.15.

  17. 17.

    When we study the interior of the moduli space \(\mathcal M_{0,\ell }(\beta )\), we can find an appropriate local trivialization of the universal family of source curves (in the sense of fiber bundle of smooth orbifolds). So there is no such issue.

  18. 18.

    It seems that under this condition one can construct a Kuranishi model of S −1(0) at each point.

  19. 19.

    Applications to the moduli space of pseudo-holomorphic curves based on such a chain model are announced but are not available yet.

  20. 20.

    [FOn2, appendix],[FOOO21] etc.

  21. 21.

    http://media.scgp.stonybrook.edu/presentations/2013/20140318_McDuff.pdf

  22. 22.

    A similar method was used by Ruan [Ru2, Lemma 2.5 (page 171)].

  23. 23.

    Liu and Tian [LiuTi, page 8, line 3] took the same choice as McDuff and Wehrheim.

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Authors and Affiliations

  1. Simons Center for Geometry and Physics, State University of New York, Stony Brook, NY, USA

    Kenji Fukaya

  2. IBS Center for Geometry and Physics Pohang, Gyung-Buk, Korea

    Yong-Geun Oh

  3. Department of Mathematics, Postech Pohang, Gyung-Buk, Korea

    Yong-Geun Oh

  4. Graduate School of Mathematics Nagoya University, Nagoya, Japan

    Hiroshi Ohta

  5. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    Kaoru Ono

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  1. Kenji Fukaya
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Fukaya, K., Oh, YG., Ohta, H., Ono, K. (2020). Introduction. In: Kuranishi Structures and Virtual Fundamental Chains. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-5562-6_1

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