Images of real hypersurfaces under holomorphic mappings
type: Item
instance of: scholarly article
title: Images of real hypersurfaces under holomorphic mappings
zbMATH Open document ID: 0770.32009
author: Mohamed Salah Baouendi
author: Linda Preiss Rothschild
published in: Journal of Differential Geometry
publication date: 1993-04-01T00:00:00Z
review text: This paper is devoted to the mapping problem for real analytic hypersurfaces in complex vector spaces. It is a continuation of the authors' investigations in J. Differ. Geom. 31, No. 2, 473-499 (1990; Zbl 0702.32014) and also of these one with \textit{S. R. Bell} in Duke Math. J. 56, No. 3, 503-530 (1988; Zbl 0655.32015). Given a germ of real analytic hypersurface \(M\) through 0 in \(\mathbb{C}^{n+I}\) and a germ \(H\) at 0 of a holomorphic mapping of \(\mathbb{C}^{n+I}\) into itself, the problem, whether there exists a real analytic hypersurface \(M'\) in \(\mathbb{C}^{n+I}\) such that \(H(M)\subset M'\), is completely solved for the case when \(M\) is essentially finite. The case when \(H(M)\) is a smooth real hypersurface is characterized for finite \(H\). Assuming that \(H(M)=M'\) is a smooth submanifold, it is proved that \(M'\) is real analytic. An example is given that the smoothness assumption for \(M\) cannot be weakened. Images of hypersurfaces under nonfinite \(H\) are studied too. There are many interesting details in the proposed proofs, as well as different comments and statements of open problems.
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32H02
zbMATH DE Number: 147457
zbMATH Keywords: real analytic hypersurface
zbMATH Keywords: essentially finite analytic hypersurface
zbMATH Keywords: finite holomorphic map
Wikidata QID: Q115180691
reviewed by: Stancho Dimiev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214448443
OpenAlex ID: W1579777984
DOI: 10.4310/JDG/1214448443
Recommended article: Images of real submanifolds under finite holomorphic mappings
Recommended article: Q3979229
Recommended article: AN ESTIMATE OF THE DIMENSION OF THE IMAGE UNDER A HOLOMORPHIC MAPPING OF REAL-ANALYTIC HYPERSURFACES
Recommended article: Q3029303
Recommended article: Extension of holomorphic mappings along real-analytic hypersurfaces
Recommended article: On some hypersurfaces and holomorphic mappings
Recommended article: Q4342673
Recommended article: Real hypersurfaces of a complex projective space in terms of holomorphic distribution
Recommended article: Q4711776
Recommended article: Holomorphic Mappings of Complex Spaces
description: scientific article; zbMATH DE number 147457