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Introduction to Part II

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Kuranishi Structures and Virtual Fundamental Chains

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Abstract

In Part I, we described the foundation of the theory of Kuranishi structures, good coordinate systems, and CF-perturbations (also multivalued perturbations), and we defined the integration along the fiber (pushout) of a strongly submersive map with respect to a CF-perturbation and also proved Stokes’ formula.

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Notes

  1. 1.

    The equivalence class is defined by using the symplectic area and the Maslov index.

  2. 2.

    Though it is better to write it as \(\overset {\circ }{\mathcal M}_{k+1}(L;\beta )\), we omit L for the simplicity of notation.

  3. 3.

    Strictly speaking, in (22.17) we define a partial A ∞ algebra structure (see Definition 21.22) which depends on a parameter 𝜖 > 0. We then use a ‘homotopy limit’ in a way similar to that of the construction of A ∞ algebra explained in Sect. 15.1.7.

  4. 4.

    Actually Hutchings’ concern is not so much about the proof of independence of Morse–Novikov cohomology of the choices but rather the explicit form of the cochain homotopy equivalence, between two Morse–Novikov complexes before and after wall crossing. The discussion below clarifies the way to prove the independence of Morse–Novikov cohomology, but to find the explicit form of the cochain homotopy equivalence we need to study more. See [H1].

  5. 5.

    Our exposition below is a slightly improved version of the one that appeared in [FOOO14]. In [FOOO4] we took a countably generated subcomplex of a smooth singular chain complex. (We use Baire’s category theorem uncountably many times in [FOOO14] so that we do not need to take a countably generated subcomplex as we did in [FOOO4].) Here we take singular chain complex itself, that is the way of [FOOO14].

  6. 6.

    The one in [Jo4] may be regarded to be closer to a ‘coordinate free’ definition. His definition of a Kuranishi structure there is different from the one in this book.

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Author information

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

Authors
  1. Kenji Fukaya
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  2. Yong-Geun Oh
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  3. Hiroshi Ohta
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  4. Kaoru Ono
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Fukaya, K., Oh, YG., Ohta, H., Ono, K. (2020). Introduction to Part II. In: Kuranishi Structures and Virtual Fundamental Chains. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-5562-6_15

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