Open Access
Spring 2012 Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions
Peter Ebenfelt, Duong Ngoc Son
Illinois J. Math. 56(1): 33-51 (Spring 2012). DOI: 10.1215/ijm/1380287458

Abstract

We consider holomorphic mappings H between a smooth real hypersurface MCn+1 and another MCN+1 with Nn. We provide conditions guaranteeing that H is transversal to M along all of M. In the strictly pseudoconvex case, this is well known and follows from the classical Hopf boundary lemma. In the equidimensional case (N=n), transversality holds for maps of full generic rank provided that the source is of finite type in view of recent results by the authors (see also a previous paper by the first author and L. Rothschild). In the positive codimensional case (N>n), the situation is more delicate as examples readily show. In recent work by S. Baouendi, the first author, and L. Rothschild, conditions were given guaranteeing that the map H is transversal outside a proper subvariety of M, and examples were given showing that transversality may fail at certain points.

One of the results in this paper implies that if N2n2, M is Levi-nondegenerate, and H has maximal rank outside a complex subvariety of codimension 2, then H is transversal to M at all points of M. We show by examples that this conclusion fails in general if N2n, or if the set WH of points where H is not of maximal rank has codimension one. We also show that is transversal at all points if is assumed to be a finite map (which allows to have codimension one) and the stronger inequality holds, provided that is of finite type.

Citation

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Peter Ebenfelt. Duong Ngoc Son. "Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions." Illinois J. Math. 56 (1) 33 - 51, Spring 2012. https://doi.org/10.1215/ijm/1380287458

Information

Published: Spring 2012
First available in Project Euclid: 27 September 2013

zbMATH: 1293.32043
MathSciNet: MR3117016
Digital Object Identifier: 10.1215/ijm/1380287458

Subjects:
Primary: 32H02 , 32V30

Rights: Copyright © 2012 University of Illinois at Urbana-Champaign

Vol.56 • No. 1 • Spring 2012
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