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Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: II

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

Abstract

This is the second of a series of two articles in which we provide detailed and self-contained account of the construction of a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks. Using the notion of obstruction bundle data introduced in [FOOO8], we give a systematic way of constructing a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks which are compatible at the boundary and corners. More specifically, it defines a tree like K-system in the sense of [FOOO6, Definition 21.9]. The method given in this paper does not only simplify the description of the constructions in the earlier literature, but also is designed to provide a systematic utility tool for the construction of a system of Kuranishi structures in the future research. We also establish its uniqueness.

Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: II

Abstract

This is the second of a series of two articles in which we provide detailed and self-contained account of the construction of a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks. Using the notion of obstruction bundle data introduced in [FOOO8], we give a systematic way of constructing a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks which are compatible at the boundary and corners. More specifically, it defines a tree like K-system in the sense of [FOOO6, Definition 21.9]. The method given in this paper does not only simplify the description of the constructions in the earlier literature, but also is designed to provide a systematic utility tool for the construction of a system of Kuranishi structures in the future research. We also establish its uniqueness.

Paper Structure

This paper contains 15 sections, 48 theorems, 166 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Obstruction bundle data: Review
  4. Stratification of the moduli space
  5. Disk-component-wise-ness of obstruction bundle data
  6. Disk-component-wise-ness implies corner compatibility condition
  7. Existence of a disk-component-wise system of obstruction bundle data 1
  8. The idea of the construction
  9. Stabilization by interior marked points
  10. Iteration of the construction of Subsection \ref{['subsec:staint']}
  11. Existence of a disk-component-wise system of obstruction bundle data 2
  12. Uniqueness of the Kuranishi structure up to pseudo isotopy
  13. The case of a single $K$-space
  14. The case of a system of $K$-spaces
  15. Independence of almost complex structure up to pseudo-isotopy

Key Result

Theorem 2.5

In Situation situ271, there exists a tree-like K-systemIn the previous literature we used the terminology "$A_{\infty}$ correspondence", which is the same as "tree-like K-system". whose moduli spaces of operations are $\mathcal{M}_{k+1}(X,L,J;\beta)$.

Figures (6)

  • Figure 1: $s({\rm e})$ and $t({\rm e})$
  • Figure 2: ${\bf p},{\bf q},{\frak p}$
  • Figure 4: Forgetting marked points $\vec{z}$, $\vec{\frak z}$
  • Figure 5: Sublemma \ref{['sublem777']}
  • Figure 6: Situation \ref{['situ69']}
  • ...and 1 more figures

Theorems & Definitions (125)

  • Remark 1.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.5
  • Corollary 2.6
  • Theorem 2.8
  • Corollary 2.9
  • Remark 2.10
  • Definition 2.11
  • Definition 2.12
  • ...and 115 more