{
  "id": "1511.05075",
  "version": "v1",
  "published": "2015-11-16T18:25:16.000Z",
  "updated": "2015-11-16T18:25:16.000Z",
  "title": "A long $\\mathbb C^2$ without holomorphic functions",
  "authors": [
    "Luka Boc Thaler",
    "Franc Forstneric"
  ],
  "categories": [
    "math.CV"
  ],
  "abstract": "For every integer $n>1$ we construct a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of $\\mathbb C^n$ (i.e., a long $\\mathbb C^n$), but it does not admit any nonconstant holomorphic functions. We also introduce new biholomorphic invariants of a complex manifold, the stable core and the strongly stable core, and prove that every compact strongly pseudoconvex polynomially convex domain $B\\subset \\mathbb C^n$ is the strongly stable core of a long $\\mathbb C^n$; in particular, non-equivalent domains give rise to non-equivalent long $\\mathbb C^n$'s. Furthermore, we show that for any open set $U\\subset \\mathbb C^n$ there exists a long $\\mathbb C^n$ whose stable core is dense in $U$. Thus, for any $n>1$ there exist uncountably many pairwise non-equivalent long $\\mathbb C^n$'s.These results answer several long standing open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-16T18:25:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32E10",
      "32E30",
      "32H02"
    ],
    "keywords": [
      "complex manifold",
      "strongly stable core",
      "non-equivalent long",
      "nonconstant holomorphic functions",
      "long standing open problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}