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

. 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.

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Ahern, Patrick, and Gong, Xianghong. "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 18.1 (2009): 1-64. <http://eudml.org/doc/10107>.

@article{Ahern2009,
abstract = {We will classify $n$-dimensional real submanifolds in $\{\{\mathbb\{C\}\}\}^n$ 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 $\{\{\mathbb\{C\}\}\}^n$ such that its images under biholomorphisms $(z_1, \dots , z_n) \mapsto (rz_1, \dots , rz_\{n-1\}, r^2z_n)$, $r > 1$, are not equivalent to $M$ via any local volume-preserving holomorphic map.},
affiliation = {Department of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A.; Department of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A.},
author = {Ahern, Patrick, Gong, Xianghong},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {CR singular points; formal equivalence; parabolic complex tangents; -dimensional real submanifolds in },
language = {eng},
month = {6},
number = {1},
pages = {1-64},
publisher = {Université Paul Sabatier, Toulouse},
title = {Real analytic manifolds in $\{\mathbb\{C\}\}^n$ with parabolic complex tangents along a submanifold of codimension one},
url = {http://eudml.org/doc/10107},
volume = {18},
year = {2009},
}

TY - JOUR
AU - Ahern, Patrick
AU - Gong, Xianghong
TI - Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2009/6//
PB - Université Paul Sabatier, Toulouse
VL - 18
IS - 1
SP - 1
EP - 64
AB - We will classify $n$-dimensional real submanifolds in ${{\mathbb{C}}}^n$ 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 ${{\mathbb{C}}}^n$ such that its images under biholomorphisms $(z_1, \dots , z_n) \mapsto (rz_1, \dots , rz_{n-1}, r^2z_n)$, $r > 1$, are not equivalent to $M$ via any local volume-preserving holomorphic map.
LA - eng
KW - CR singular points; formal equivalence; parabolic complex tangents; -dimensional real submanifolds in
UR - http://eudml.org/doc/10107
ER -

References

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Voronin (S.M.).— The Darboux-Whitney theorem and related questions, in Nonlinear Stokes phenomena, p. 139–233, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, (1993). Zbl0789.58015 MR1206044
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