A geometric approach to Catlin’s boundary systems

Zaitsev, Dmitri

doi:10.5802/aif.3304

A geometric approach to Catlin’s boundary systems [ Une approche géométrique aux systèmes bord de Catlin ]

Zaitsev, Dmitri

Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2635-2679.

Résumé
Abstract

Soit $p$ un point d’une hypersurface $M \subset ℂ^{n},$ réelle et lisse, en lequel la forme de Levi possède un noyau non trivial, noté $K_{p}^{10} \neq 0$ . On introduit alors un tenseur d’ordre $3,$ invariant

τ_{p}^{3} : ℂ T_{p} \times K_{p}^{10} \times \bar{K_{p}^{10}} \to ℂ \otimes (T_{p} / H_{p}),

qui, avec le tenseur d’ordre $3$ d’Ebenfelt, fournit l’ensemble de tous les tenseurs d’ordre $3$ invariants de $M$ au point $p$ .

Si, de plus, on suppose que $M$ est pseudo-convexe, alors $τ_{p}^{3}$ est identiquement nul. On définit alors un tenseur d’ordre $4$ , invariant

τ_{p}^{4} : ℂ T_{p} \times ℂ T_{p} \times K_{p}^{10} \times \bar{K_{p}^{10}} \to ℂ \otimes (T_{p} / H_{p}) .

On introduit également pour $q = 0, ..., n - 1$ , un faisceau de sous-modules invariant, noté $𝒮^{10} (q)$ , constitué de champs de vecteurs de type $(1, 0)$ , construits à partir de la forme de Levi, ainsi qu’un faisceau d’idéaux invariant, noté $ℐ (q)$ , 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 $q$ est alors contenu localement dans des sous-variétés réelles et lisses définies par des parties rélles de fonctions dans $ℐ (q)$ , et dont les espaces tangents ont des descriptions algébriques explicites en termes du tenseur $τ^{4}$ .

Tout récemment, la construction de $τ^{3}$ et $τ^{4}$ a inspiré Raich et Harrington [35] qui ont découvert de nouveaux invariants associés à ces derniers dans le cas où $M$ n’est pas pseudo-convexe et les ont reliés à la « closed range property » pour l’opérateur $\bar{\partial}$ .

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 $p$ in a smooth real hypersurface $M \subset ℂ^{n}$ , where the Levi form has the nontrivial kernel $K_{p}^{10} \neq 0$ , we introduce an invariant cubic tensor

τ_{p}^{3} : ℂ T_{p} \times K_{p}^{10} \times \bar{K_{p}^{10}} \to ℂ \otimes (T_{p} / H_{p}),

which together with Ebenfelt’s $3$ rd order tensor, constitutes the full set of the $3$ rd order invariants of $M$ at $p$ .

Next, in addition, assume $M \subset ℂ^{n}$ to be (weakly) pseudoconvex. Then $τ_{p}^{3}$ must identically vanish. In this case we further define an invariant quartic tensor

τ_{p}^{4} : ℂ T_{p} \times ℂ T_{p} \times K_{p}^{10} \times \bar{K_{p}^{10}} \to ℂ \otimes (T_{p} / H_{p}),

and for every $q = 0, \dots, n - 1$ , an invariant submodule sheaf $𝒮^{10} (q)$ of $(1, 0)$ vector fields in terms of the Levi form, and an invariant ideal sheaf $ℐ (q)$ of complex functions generated by certain components and derivatives of the Levi form, such that the set of points of Levi rank $q$ is locally contained in real smooth submanifolds defined by real parts of the functions in $ℐ (q)$ , whose tangent spaces have explicit algebraic description in terms of the quartic tensor $τ^{4}$ .

Most recently, the constructions of $τ^{3}$ and $τ^{4}$ 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 $\bar{\partial}$ operator.

Finally, we relate the introduced invariants with D’Angelo’s finite type, Catlin’s multitype and Catlin’s boundary systems.

Reçu le : 2018-01-22
Accepté le : 2018-07-12
Publié le : 2019-10-29

DOI : https://doi.org/10.5802/aif.3304
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

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     author = {Zaitsev, Dmitri},
     title = {A geometric approach to Catlin's boundary systems},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {6},
     year = {2019},
     pages = {2635-2679},
     doi = {10.5802/aif.3304},
     language = {en},
     url = {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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