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

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

Abstract

This is the first 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 exponential decay estimate given in [FOOO7]. This article completes the construction of a Kuranishi structure of a single moduli space. This article is an improved version of [FOOO4, Part 4] and its mathematical content is taken from our earlier writing [FOn,FOOO2,FOOO4,FOOO7]. The version 2 contains a correction of typological errors in the statement of Lemma 10.11 and the statement of Proposition 10.4. The proof of them are not changed. (Note these places were written in the same way as version 1 in the version published in Surveys in Differential Geometry.)

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

Abstract

This is the first 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 exponential decay estimate given in [FOOO7]. This article completes the construction of a Kuranishi structure of a single moduli space. This article is an improved version of [FOOO4, Part 4] and its mathematical content is taken from our earlier writing [FOn,FOOO2,FOOO4,FOOO7]. The version 2 contains a correction of typological errors in the statement of Lemma 10.11 and the statement of Proposition 10.4. The proof of them are not changed. (Note these places were written in the same way as version 1 in the version published in Surveys in Differential Geometry.)

Paper Structure

This paper contains 40 sections, 44 theorems, 131 equations, 3 figures.

Table of Contents

  1. Statement of the results
  2. Universal family of marked disks and spheres
  3. Analytic family of coordinates at the marked points and local trivialization of the universal family
  4. Analytic family of coordinates at the marked points
  5. Analytic families of coordinates and complex/smooth structure of the moduli space
  6. Local trivialization of the universal family
  7. Stable map topology and $\epsilon$-closeness
  8. Partial topology
  9. Weak stabilization data
  10. The $\epsilon$-closeness
  11. Obstruction bundle data
  12. Kuranishi structure: review
  13. Construction of Kuranishi structure
  14. Statement
  15. Construction of Kuranishi charts
  16. ...and 25 more sections

Key Result

Theorem 1.1

$\mathcal{M}_{k+1,\ell}(X,L,J;\beta)$ carries a Kuranishi structure with corners.

Figures (3)

  • Figure 1: The map $\Phi$
  • Figure 2: ${\bf p}$,${\bf q}$ and ${\bf x}$
  • Figure 3: ${\bf p}^+$ and ${\bf q}^+$

Theorems & Definitions (136)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 3.1
  • Definition 3.2
  • ...and 126 more