Stratifications associated to generic closed two-forms and stratified $L_\infty$ spaces
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| Format: | Preprint |
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2026
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| author | Kim, Taesu Oh, Yong-Geun |
| author_facet | Kim, Taesu Oh, Yong-Geun |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_24099 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stratifications associated to generic closed two-forms and stratified $L_\infty$ spaces Kim, Taesu Oh, Yong-Geun Symplectic Geometry Differential Geometry Primary 53D42, Secondary 58J32 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. |
| title | Stratifications associated to generic closed two-forms and stratified $L_\infty$ spaces |
| topic | Symplectic Geometry Differential Geometry Primary 53D42, Secondary 58J32 |
| url | https://arxiv.org/abs/2602.24099 |