Géométrie différentielle, Systèmes dynamiques
Deformation of singular foliations, 1: Local deformation cohomology
Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 273-283.

In this paper we introduce the notion of deformation cohomology for singular foliations and related objects (namely integrable differential forms and Nambu structures), and study it in the local case, i.e., in the neighborhood of a point.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.26
Monnier, Philippe 1 ; Nguyen, Tien Zung 1

1 Institut de Mathématiques de Toulouse, UMR 5219 CNRS, Université Toulouse III, France
@article{CRMATH_2020__358_3_273_0,
     author = {Monnier, Philippe and Nguyen, Tien Zung},
     title = {Deformation of singular foliations, 1: {Local} deformation cohomology},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {273--283},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {3},
     year = {2020},
     doi = {10.5802/crmath.26},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.26/}
}
TY  - JOUR
AU  - Monnier, Philippe
AU  - Nguyen, Tien Zung
TI  - Deformation of singular foliations, 1: Local deformation cohomology
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 273
EP  - 283
VL  - 358
IS  - 3
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.26/
DO  - 10.5802/crmath.26
LA  - en
ID  - CRMATH_2020__358_3_273_0
ER  - 
%0 Journal Article
%A Monnier, Philippe
%A Nguyen, Tien Zung
%T Deformation of singular foliations, 1: Local deformation cohomology
%J Comptes Rendus. Mathématique
%D 2020
%P 273-283
%V 358
%N 3
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.26/
%R 10.5802/crmath.26
%G en
%F CRMATH_2020__358_3_273_0
Monnier, Philippe; Nguyen, Tien Zung. Deformation of singular foliations, 1: Local deformation cohomology. Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 273-283. doi : 10.5802/crmath.26. http://www.numdam.org/articles/10.5802/crmath.26/

[1] Androulidakis, Iakovos; Skandalis, Georges The holonomy groupoid of a singular foliation, J. Reine Angew. Math., Volume 626 (2009), pp. 1-37 | DOI | MR | Zbl

[2] Androulidakis, Iakovos; Zambon, Marco Stefan–Sussmann singular foliations, singular subalgebroids and their associated sheaves, Int. J. Geom. Methods Mod. Phys., Volume 13 (2016), 1641001, 17 pages | MR | Zbl

[3] Arnold, Vladimir I.; Gusein-Zade, Sabir M.; Varchenko, Alexander N. Singularities of differentiable maps. Volume I: The classification of critical points, caustics and wave fronts, Monographs in Mathematics, 82, Birkhäuser, 1985 | Zbl

[4] Dufour, Jean-Paul; Zung, Nguyen Tien Linearization of Nambu structures, Compos. Math., Volume 117 (1999) no. 1, p. 77--98 | MR | Zbl

[5] Dufour, Jean-Paul; Zung, Nguyen Tien Poisson structures and their normal forms, Progress in Mathematics, 242, Birkhäuser, 2005 | MR | Zbl

[6] Heitsch, James L. A cohomology for foliated manifolds, Comment. Math. Helv., Volume 50 (1975), pp. 197-218 | DOI | MR | Zbl

[7] Hermann, Robert On the accessibility problem in control theory, International symposium on nonlinear differential equations and nonlinear mechanics, Academic Press Inc., 1963, pp. 327-332 | Zbl

[8] Malgrange, Bernard Frobenius avec singularités. II. Le cas général, Invent. Math., Volume 39 (1977) no. 1, pp. 67-89 | DOI | Zbl

[9] Minh, Truong Hong; Zung, Nguyen Tien Commuting Foliations, Regul. Chaotic Dyn., Volume 18 (2013) no. 6, pp. 608-622 | MR | Zbl

[10] Monnier, Philippe Computations of Nambu-Poisson cohomologies, Int. J. Math., Volume 26 (2001) no. 2, pp. 65-81 | DOI | MR | Zbl

[11] Monnier, Philippe Poisson cohomology in dimension two, Isr. J. Math., Volume 129 (2002), pp. 189-207 | DOI | MR | Zbl

[12] Reeb, Georges Sur les espaces fibrés et les variétés feuilletées. II: Sur certaines propriétés topologiques des variétés feuilletées (Actualités scientifiques et industrielles), Volume 1183, Hermann & Cie, 1952, p. 5-89, 155–156 | Zbl

[13] Stefan, Peter Accessible sets, orbits, and foliations with singularities, Proc. Lond. Math. Soc., Volume 29 (1974), pp. 699-713 | DOI | MR | Zbl

[14] Sussmann, Hector J. Orbits of families of vector fields and integrability of distributions, Trans. Am. Math. Soc., Volume 180 (1973), pp. 171-188 | DOI | MR | Zbl

[15] Thurston, William P. A generalization of the Reeb stability theorem, Topology, Volume 13 (1974), pp. 347-352 | DOI | MR | Zbl

[16] Zung, Nguyen Tien New results on the linearization of Nambu structures, J. Math. Pures Appl., Volume 99 (2013) no. 2, pp. 211-218 | DOI | MR | Zbl

Cité par Sources :