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Stratifications associated to generic closed two-forms and stratified $L_\infty$ spaces

Taesu Kim, Yong-Geun Oh

Abstract

Jae-Suk Park and the second-named author introduce the deformation problem of coisotropic submanifolds of a symplectic manifold as the study of Mauer-Cartan moduli problem of an $L_\infty$ algebra attached to the foliation de-Rham complex associated to the null foliation of the corresponding presymplectic structure. The main purpose of the present paper is to extend this study of $L_\infty$ structures to the case of generic closed two-forms on arbitrary smooth manifolds as a stratified $L_\infty$ space. We first prove that there exists a residual subset of closed 2-forms, which we denote by $Z^2_{reg}(M) \subset Z^2(M)$, such that any element $ω$ therefrom admits a Whitney stratification each of whose strata is a presymplectic manifold. We then associate an $L_\infty$ space to each stratum (and to its tubular neighborhood) and glue the collection of $L_\infty$ spaces to a global stratified $L_\infty$ space by the coordinate atlas consisting of $L_\infty$ morphisms, which is a collection of $L_\infty$ morphisms, not necessarily of quasi-isomorphisms.

Stratifications associated to generic closed two-forms and stratified $L_\infty$ spaces

Abstract

Jae-Suk Park and the second-named author introduce the deformation problem of coisotropic submanifolds of a symplectic manifold as the study of Mauer-Cartan moduli problem of an algebra attached to the foliation de-Rham complex associated to the null foliation of the corresponding presymplectic structure. The main purpose of the present paper is to extend this study of structures to the case of generic closed two-forms on arbitrary smooth manifolds as a stratified space. We first prove that there exists a residual subset of closed 2-forms, which we denote by , such that any element therefrom admits a Whitney stratification each of whose strata is a presymplectic manifold. We then associate an space to each stratum (and to its tubular neighborhood) and glue the collection of spaces to a global stratified space by the coordinate atlas consisting of morphisms, which is a collection of morphisms, not necessarily of quasi-isomorphisms.
Paper Structure (37 sections, 38 theorems, 259 equations)

This paper contains 37 sections, 38 theorems, 259 equations.

Table of Contents

  1. Introduction and overview
  2. Motivation
  3. Statement of main results
  4. Discussion
  5. $L_\infty$ spaces and presymplectic manifolds: Revisit
  6. Curved $L_\infty$ spaces and tangent complexes
  7. The case of presymplectic manifolds
  8. Description via $V$-algebra and Maurer-Cartan elements
  9. Derivation of $L_\infty$ structure from $V$-algebra
  10. Formal deformation of coisotropic submanifolds
  11. Presymplectic stratification of a generic closed two-form
  12. Stratification of the set of skew-symmetric bilinear forms
  13. Stratawise transversality of generic closed two-forms
  14. Off-shell framework for the transversality study
  15. Parametrization of closed two-forms
  16. ...and 22 more sections

Key Result

Theorem 1.4

There exists a residual subset, denoted by $Z^2_{\text{\rm reg}}(M)$, of $Z^2(M)$ such that the decomposition ${\mathcal{S}}_\omega = \{Y_m(\omega)\}_{m=0}^{N}$ forms a Whitney stratification of $M$ for any $\omega \in Z^2_{\text{\rm reg}}(M)$.

Theorems & Definitions (80)

  • Remark 1.3
  • Theorem 1.4: Theorem \ref{['thm:adaptedness']}
  • Definition 1.5: Polarized presymplectic manifolds
  • Theorem 1.6: Theorem \ref{['thm:gluing-morphisms']}
  • Definition 2.1: Virtual dimension
  • Theorem 2.2: Theorem 9.4 oh-park:coisotropic
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 70 more