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$L_\infty$-Kuranishi spaces and the moduli space of pseudoholomorphic maps

Taesu Kim

Abstract

We introduce $L_{\infty}$-Kuranishi spaces by associating, to each chart, $L_{\infty}[1]$-algebras defined on open neighborhoods of the zero points of the Kuranishi section. We show that these objects collectively form a category, which naturally embeds the category of smooth manifolds. Certain notions in \cite{FOOO1} are modified to achieve desired categorical structures; for instance, the tangent bundle condition is interpreted as a quasi-isomorphism condition for the $L_{\infty}$-structures. In this process, the originally strict and rigid cocycle condition for coordinate changes is replaced by more flexible homotopy-theoretic compatibilities. To this end, a model of higher homotopy theory for $L_{\infty}[1]$-morphisms is proposed. Moreover, the moduli space of pseudoholomorphic disks with Lagrangian boundary condition is shown to serve as an example of $L_{\infty}$-Kuranishi spaces, provided that a Whitney stratification with a compatible system of tubular neighborhoods exists on each chart. Finally, the forgetful and evaluation maps for the moduli space are lifted to morphisms between $L_{\infty}$-Kuranishi spaces.

$L_\infty$-Kuranishi spaces and the moduli space of pseudoholomorphic maps

Abstract

We introduce -Kuranishi spaces by associating, to each chart, -algebras defined on open neighborhoods of the zero points of the Kuranishi section. We show that these objects collectively form a category, which naturally embeds the category of smooth manifolds. Certain notions in \cite{FOOO1} are modified to achieve desired categorical structures; for instance, the tangent bundle condition is interpreted as a quasi-isomorphism condition for the -structures. In this process, the originally strict and rigid cocycle condition for coordinate changes is replaced by more flexible homotopy-theoretic compatibilities. To this end, a model of higher homotopy theory for -morphisms is proposed. Moreover, the moduli space of pseudoholomorphic disks with Lagrangian boundary condition is shown to serve as an example of -Kuranishi spaces, provided that a Whitney stratification with a compatible system of tubular neighborhoods exists on each chart. Finally, the forgetful and evaluation maps for the moduli space are lifted to morphisms between -Kuranishi spaces.

Paper Structure

This paper contains 54 sections, 82 theorems, 551 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $L_{\infty}$-Kuranishi charts
  3. Morphisms of charts and relation to the FOOO's works
  4. Definition of $L_{\infty}$-Kuranishi spaces via higher homotopies
  5. Categorical structures
  6. The moduli space example
  7. Outline of the paper
  8. Homotopy theory for $L_\infty[1]$-algebras
  9. Homotopies of $L_{\infty}[1]$-morphisms
  10. Homotopies of $L_\infty[1]$-morphisms
  11. Homotopy equivalence of $L_\infty[1]$-algebras
  12. Whitehead theorem
  13. Higher homotopies of $L_{\infty}[1]$-morphisms
  14. Models of $\Delta^n \times C$ with $n \geq 2$
  15. Proof of Proposition \ref{['pphhe']}
  16. ...and 39 more sections

Key Result

Theorem 1.1

An FOOO's embedding of Kuranishi charts (with some more natural conditions) determines an embedding in our sense.

Figures (1)

  • Figure 1: 4 types of the nullity graphs over [$t_i, t_{i+1}$]

Theorems & Definitions (234)

  • Theorem 1.1
  • Theorem 1.3: Existence of filling homotopies for quasi-isomorphisms
  • Theorem 1.4: Higher cocycle conditions
  • Theorem 1.5
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1: Models of $\Delta^1 \times C$
  • Remark 2.2
  • Definition 2.3: Homotopy
  • Lemma 2.4
  • ...and 224 more