Local reduction theorems and invariants for singular contact structures
Annales de l'Institut Fourier, Volume 51 (2001) no. 1, p. 237-295

A differential 1-form on a $\left(2k+1\right)$-dimensional manifolds $M$ defines a singular contact structure if the set $S$ of points where the contact condition is not satisfied, $S=\left\{p\in M:\left(\omega \wedge {\left(d\omega \right)}^{k}\left(p\right)=0\right\}$, is nowhere dense in $M$. Then $S$ is a hypersurface with singularities and the restriction of $\omega$ to $S$ can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation $\left(\omega \right)$ generated by $\omega$ is determined, up to a diffeomorphism, by its restriction to $S$, if we eliminate certain degenerated singularities of $\omega$ (in the holomorphic case they form a set of infinite codimension). We also define other invariants of local singular contact structures: orientations, a line bundle, and a partial connection. We study the problem when these invariants, together with the hypersurface $S$ and the restriction of the Pfaffian equation $\left(\omega \right)$ to $S$, form a complete set of local invariants. Our results include complete solutions to this problem in dimension 3 and in the case where $S$ has no singularities.

Soit $\omega$ une 1-forme différentielle locale sur une variété $M$ de dimension $2k+1$. Par définition, elle définit une structure locale singulière de contact si le lieu $S$ de ses points singuliers $S=\left\{p\in M:\left(\omega \wedge {\left(d\omega \right)}^{k}\right)\left(p\right)=0\right\}$ est nulle part dense. Dans un tel cas on peut définir la restriction (pullback) ${\omega |}_{S}$ de $\omega$ sur l’hypersurface singulière $S$. Nos théorèmes disent que, dans les catégories holomorphe, analytique réelle et ${C}^{\infty }$, l’équation locale de Pfaff ${\omega |}_{S}=0$ sur $S$ détermine l’équation locale de Pfaff $\omega =0$ sur $M$, à un difféomorphisme près, si on exclut certaines dégénérescences de codimension infinie de $\omega$. De plus, si $S$ est lisse, l’équation locale de Pfaff $\omega =0$ sur $M$ est déterminée, à un difféomorphisme près, par sa restriction sur $S$ et deux invariants complémentaires: une orientation et une connexion partielle. Ces invariants sont en général indépendants. Nos résultats impliquent une classification des singularités des équations de Pfaff locales en dimension 3.

DOI : https://doi.org/10.5802/aif.1823
Classification:  58A17,  53B99
Keywords: contact structure, singularity, pfaffian equation, equivalence, local invariants, reduction theorems, homotopy method
@article{AIF_2001__51_1_237_0,
author = {Jakubczyk, Bronislaw and Zhitomirskii, Michail},
title = {Local reduction theorems and invariants for singular contact structures},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {51},
number = {1},
year = {2001},
pages = {237-295},
doi = {10.5802/aif.1823},
zbl = {1047.53051},
mrnumber = {1821076},
language = {en},
url = {http://www.numdam.org/item/AIF_2001__51_1_237_0}
}

Jakubczyk, Bronislaw; Zhitomirskii, Michail. Local reduction theorems and invariants for singular contact structures. Annales de l'Institut Fourier, Volume 51 (2001) no. 1, pp. 237-295. doi : 10.5802/aif.1823. http://www.numdam.org/item/AIF_2001__51_1_237_0/

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