Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one

Patrick Ahern; Xianghong Gong

doi:10.5802/afst.1204

Real analytic manifolds in

ℂ^{n}

with parabolic complex tangents along a submanifold of codimension one

Patrick Ahern ¹; Xianghong Gong ¹

¹ Department of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 1, pp. 1-64.

Abstract
Abstract

We will classify $n$ -dimensional real submanifolds in $ℂ^{n}$ which have a set of parabolic complex tangents of real dimension $n - 1$ . All such submanifolds are equivalent under formal biholomorphisms. We will show that the equivalence classes under convergent local biholomorphisms form a moduli space of infinite dimension. We will also show that there exists an $n$ -dimensional submanifold $M$ in $ℂ^{n}$ such that its images under biholomorphisms $(z_{1}, \dots, z_{n}) \mapsto (r z_{1}, \dots, r z_{n - 1}, r^{2} z_{n})$ , $r > 1$ , are not equivalent to $M$ via any local volume-preserving holomorphic map.

Nous classifions les sous-variétés réelles analytiques de dimension $n$ dans $ℂ^{n}$ , qui ont un ensemble de points de tangence complexe paraboliques de dimension réelle $n - 1$ . Ces sous variétés sont toutes équivalentes via biholomorphisme formel. Nous montrons que les classes d’équivalence sous changement de variables par biholomorphisme local (convergent) forment un ’espace de modules’ de dimension infinie. Nous montrons aussi qu’il existe une sous-variété $M$ de dimension $n$ dans $ℂ^{n}$ , dont les images par les biholomorphismes $(z_{1}, \dots, z_{n}) \mapsto (r z_{1}, \dots, r z_{n - 1}, r^{2} z_{n})$ , $r > 1$ , ne sont pas équivalentes à $M$ via biholomorphisme local préservant le volume.

MR Zbl

DOI: 10.5802/afst.1204

Author's affiliations:

Patrick Ahern ¹; Xianghong Gong ¹

¹ Department of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A.

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     author = {Patrick Ahern and Xianghong Gong},
     title = {Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1--64},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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     volume = {Ser. 6, 18},
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Patrick Ahern; Xianghong Gong. Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 1, pp. 1-64. doi : 10.5802/afst.1204. https://afst.centre-mersenne.org/articles/10.5802/afst.1204/

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