Local reduction theorems and invariants for singular contact structures
[Théorèmes de réduction et invariants locaux des structures singulières de contact]
Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 237-295.

Soit ω 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={pM:(ω(dω) k )(p)=0} est nulle part dense. Dans un tel cas on peut définir la restriction (pullback) ω| S de ω sur l’hypersurface singulière S. Nos théorèmes disent que, dans les catégories holomorphe, analytique réelle et C , l’équation locale de Pfaff ω| S =0 sur S détermine l’équation locale de Pfaff ω=0 sur M, à un difféomorphisme près, si on exclut certaines dégénérescences de codimension infinie de ω. De plus, si S est lisse, l’équation locale de Pfaff ω=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.

A differential 1-form on a (2k+1)-dimensional manifolds M defines a singular contact structure if the set S of points where the contact condition is not satisfied, S={pM:(ω(dω) k (p)=0}, is nowhere dense in M. Then S is a hypersurface with singularities and the restriction of ω to S can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation (ω) generated by ω is determined, up to a diffeomorphism, by its restriction to S, if we eliminate certain degenerated singularities of ω (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 (ω) 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.

DOI : 10.5802/aif.1823
Classification : 58A17, 53B99
Keywords: contact structure, singularity, pfaffian equation, equivalence, local invariants, reduction theorems, homotopy method
Mot clés : structure de contact, singularité, équation de Pfaff, équivalence, invariants locaux, théorèmes de réduction, méthode homotopique
Jakubczyk, Bronislaw 1 ; Zhitomirskii, Michail 2

1 Polish Academy of Sciences, Institute of Mathematics, Sniadeckich 8, 00-950 Warsaw (Pologne)
2 Technion, Department of Mathematics, 32000 Haifa (Israël)
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Jakubczyk, Bronislaw; Zhitomirskii, Michail. Local reduction theorems and invariants for singular contact structures. Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 237-295. doi : 10.5802/aif.1823. http://www.numdam.org/articles/10.5802/aif.1823/

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