Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions

TL;DR

This paper investigates conditions under which holomorphic mappings between real hypersurfaces in complex spaces of different dimensions are transversal, extending known results and providing new criteria especially in positive codimension cases.

Contribution

It establishes new sufficient conditions for transversality of holomorphic maps between hypersurfaces, especially when the target dimension is up to twice the source dimension minus two.

Findings

Transversality holds if N ≤ 2n-2, M' is Levi-nondegenerate, and H has maximal rank outside a codimension 2 subvariety.

Counterexamples show transversality can fail if N ≥ 2n or if the set where H is not of maximal rank has codimension one.

H is transversal if H is finite and N ≤ 2n-3, assuming M is of finite type.

Abstract

We consider holomorphic mappings $H$ between a smooth real hypersurface $M\subset \bC^{n+1}$ and another $M'\subset \bC^{N+1}$ with $N\geq n$. 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 $N\le 2n-2$, $M'$ 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 $N\geq 2n$, or if the set $W_H$ of points where $H$ is not of maximal rank has codimension one. We also show that $H$ is transversal at all points if $H$ is assumed to be a finite map (which allows $W_H$ to have codimension one) and the stronger inequality $N\leq 2n-3$ holds, provided that $M$ is of finite type.