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

TL;DR

This paper extends the study of $L_inf$ structures from coisotropic submanifolds to generic closed 2-forms on manifolds, constructing stratified $L_inf$ spaces via Whitney stratifications and local $L_inf$ models.

Contribution

It introduces a stratified $L_inf$ space framework for generic closed 2-forms, generalizing previous coisotropic submanifold deformation theories.

Findings

Existence of a residual subset of closed 2-forms with Whitney stratifications

Construction of local $L_inf$ spaces for each stratum

Gluing of local $L_inf$ spaces into a global stratified structure

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 $\omega$ 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.