Soit un point d’une hypersurface réelle et lisse, en lequel la forme de Levi possède un noyau non trivial, noté . On introduit alors un tenseur d’ordre invariant
qui, avec le tenseur d’ordre d’Ebenfelt, fournit l’ensemble de tous les tenseurs d’ordre invariants de au point .
Si, de plus, on suppose que est pseudo-convexe, alors est identiquement nul. On définit alors un tenseur d’ordre , invariant
On introduit également pour , un faisceau de sous-modules invariant, noté , constitué de champs de vecteurs de type , construits à partir de la forme de Levi, ainsi qu’un faisceau d’idéaux invariant, noté , constitué de fonctions complexes construites à partir de certaines composantes et dérivées de la forme de Levi. L’ensemble des points dont la forme de Levi est de rang est alors contenu localement dans des sous-variétés réelles et lisses définies par des parties rélles de fonctions dans , et dont les espaces tangents ont des descriptions algébriques explicites en termes du tenseur .
Tout récemment, la construction de et a inspiré Raich et Harrington [35] qui ont découvert de nouveaux invariants associés à ces derniers dans le cas où n’est pas pseudo-convexe et les ont reliés à la « closed range property » pour l’opérateur .
Enfin, on fait le lien entre ces invariants définis plus haut et le type fini au sens de D’Angelo, le multitype au sens de Catlin et les « boundary systems » de Catlin.
For a point in a smooth real hypersurface , where the Levi form has the nontrivial kernel , we introduce an invariant cubic tensor
which together with Ebenfelt’s rd order tensor, constitutes the full set of the rd order invariants of at .
Next, in addition, assume to be (weakly) pseudoconvex. Then must identically vanish. In this case we further define an invariant quartic tensor
and for every , an invariant submodule sheaf of vector fields in terms of the Levi form, and an invariant ideal sheaf of complex functions generated by certain components and derivatives of the Levi form, such that the set of points of Levi rank is locally contained in real smooth submanifolds defined by real parts of the functions in , whose tangent spaces have explicit algebraic description in terms of the quartic tensor .
Most recently, the constructions of and inspired Raich and Harrington [35] to discover new related invariants in the non-pseudoconvex case and connect them with the closed range property for the operator.
Finally, we relate the introduced invariants with D’Angelo’s finite type, Catlin’s multitype and Catlin’s boundary systems.
Accepté le :
Publié le :
Classification : 32T25, 32T27, 32V05, 32V15, 32V35, 32W05, 32S60, 58K50
Mots clés : multitype de Catlin, estimée sous-elliptique, système bord, forme de Levi, pseudo-convexité, hypersurface réelle, tenseur invariant, faisceau idéal
@article{AIF_2019__69_6_2635_0, author = {Zaitsev, Dmitri}, title = {A geometric approach to Catlin's boundary systems}, journal = {Annales de l'Institut Fourier}, pages = {2635--2679}, publisher = {Association des Annales de l'institut Fourier}, volume = {69}, number = {6}, year = {2019}, doi = {10.5802/aif.3304}, language = {en}, url = {https://aif.centre-mersenne.org/item/AIF_2019__69_6_2635_0/} }
Zaitsev, Dmitri. A geometric approach to Catlin’s boundary systems. Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2635-2679. doi : 10.5802/aif.3304. https://aif.centre-mersenne.org/item/AIF_2019__69_6_2635_0/
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