{
  "id": "1106.4447",
  "version": "v1",
  "published": "2011-06-22T13:40:32.000Z",
  "updated": "2011-06-22T13:40:32.000Z",
  "title": "Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions",
  "authors": [
    "Peter Ebenfelt",
    "Duong Ngoc Son"
  ],
  "categories": [
    "math.CV"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-06-22T13:40:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32H02",
      "32V30"
    ],
    "keywords": [
      "holomorphic mappings",
      "complex spaces",
      "transversality",
      "maximal rank",
      "first author"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.4447E"
    }
  }
}