Preimages of CR submanifolds with generic holomorphic maps
type: Item
instance of: scholarly article
title: Preimages of CR submanifolds with generic holomorphic maps
zbMATH Open document ID: 1187.32028
author: Dejan Kolarič
published in: Journal d'Analyse Mathématique
publication date: 2009-12-14T00:00:00Z
review text: Let \(M \subset \mathbb{C}^{m}\) be a smooth \(CR\) manifold of codimension \(d\) and of \(CR\) dimension \(d_{1}\), let \(f :\mathbb{C}^{n} \rightarrow \mathbb{C}^{m}\) be a holomorphic map, let \(M' = f^{-1}(M)\) and denote by \(\Sigma'\) the set of points \(x \in M'\) where (the germ of) \(f\) is not \(CR\) transversal to \(M\). If \(f\) is generic, the author investigates the structure of \(\Sigma '\) and shows that it is a smooth subvariety of \(\mathbb{C}^{n}\) of real codimension \( 2(n-m+d_{1} +1)+d\) if \(n+d_{1} \geq m\) or it equals \(M'\) if \(n+d_{1} < m\). Moreover, he describes an interesting Whitney stratification of \(\Sigma '\). Further, he describes the structure of \(M'\) by analyzing what happens in \(M' \setminus \Sigma'\) and in each subset of the stratification of \(\Sigma '\). He also shows that if \(n+d_{1} \geq m\) and \( 2(n-m+d_{1} +1)+d>n\) then a generic holomorphic map is \(CR\) transversal to \(M\). Preliminary statements, necessary for the understanding of the subject are contained in the paper. Interesting examples and comments are also presented.
reviewed by: Eugen Pascu
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 32V10
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 32E10
zbMATH DE Number: 5646619
zbMATH Keywords: \(CR\)-manifold
zbMATH Keywords: generic holomorphic function
zbMATH Keywords: \(CR\) transversality
zbMATH Keywords: stratification
zbMATH Keywords: Whitney stratification
MaRDI profile type: Publication
full work available at URL: s11854-009-0010-7
OpenAlex ID: W2014961006
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DOI: 10.1007/S11854-009-0010-7
description: scientific article; zbMATH DE number 5646619