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2001 Analytic discs in symplectic spaces
Luca Baracco, Giuseppe Zampieri
Nagoya Math. J. 161: 55-67 (2001).

Abstract

We develop some symplectic techniques to control the behavior under symplectic transformation of analytic discs $A$ of $X = \mathbb{C}^{n}$ tangent to a real generic submanifold $R$ and contained in a wedge with edge $R$.

We show that if $A^{*}$ is a lift of $A$ to $T^{*}X$ and if $\chi$ is a symplectic transformation between neighborhoods of $p_{o}$ and $q_{o}$, then $A$ is orthogonal to $p_{o}$ if and only if $\widetilde{A} := \pi\chi A^{*}$ is orthogonal to $q_{o}$. Also we give the (real) canonical form of the couples of hypersurfaces of $\mathbb{R}^{2n} \simeq \mathbb{C}^{n}$ whose conormal bundles have clean intersection. This generalizes [10] to general dimension of intersection.

Combining this result with the quantized action on sheaves of the "tuboidal" symplectic transformation, we show the following: If $R$, $S$ are submanifolds of $X$ with $R \subset S$ and $p_{o} \in T^{*}_{S}X|_{R}$ but $ip_{o} \notin T^{*}_{R}X$, then the conditions $\text{cod}_{T^{\mathbb{C}}S}(T^{\mathbb{C}}R)=\text{cod}_{TS}(TR) (\text{resp. cod}_{T^{\mathbb{C}}S}(T^{\mathbb{C}}R) = 0)$ can be characterized as opposite inclusions for the couple of closed half-spaces with conormal bundles $\chi(T^{*}_{R}X)$ and $\chi(T^{*}_{S}X)$ at $\chi(p_{o})$.

In $\S$3 we give some partial applications of the above result to the analytic hypoellipticity of $CR$ hyperfunctions on higher codimensional manifolds by the aid of discs (cf. [2], [3] as for the case of hypersurfaces).

Citation

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Luca Baracco. Giuseppe Zampieri. "Analytic discs in symplectic spaces." Nagoya Math. J. 161 55 - 67, 2001.

Information

Published: 2001
First available in Project Euclid: 27 April 2005

zbMATH: 0998.53053
MathSciNet: MR1820212

Subjects:
Primary: 32V25
Secondary: 32A45

Rights: Copyright © 2001 Editorial Board, Nagoya Mathematical Journal

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