{
  "id": "1411.3384",
  "version": "v1",
  "published": "2014-11-12T22:39:18.000Z",
  "updated": "2014-11-12T22:39:18.000Z",
  "title": "Varieties of general type with the same Betti numbers as $\\mathbb P^1\\times \\mathbb P^1\\times\\ldots\\times \\mathbb P^1$",
  "authors": [
    "Amir Džambić"
  ],
  "comment": "15 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "We study quotients $\\Gamma\\backslash \\mathbb H^n$ of the $n$-fold product of the upper half plane $\\mathbb H$ by irreducible and torsion-free lattices $\\Gamma < PSL_2(\\mathbb R)^n$ with the same Betti numbers as the $n$-fold product $(\\mathbb P^1)^n$ of projective lines. Such varieties are called fake products of projective lines or fake $(\\mathbb P^1)^n$. These are higher dimensional analogs of fake quadrics. In this paper we show that the number of fake $(\\mathbb P^1)^n$ is finite (independently of $n$), we give examples of fake $(\\mathbb P^1)^4$ and show that for $n>4$ there are no fake $(\\mathbb P^1)^n$ of the form $\\Gamma\\backslash \\mathbb H^n$ with $\\Gamma$ contained in the norm-1 group of a maximal order of a quaternion algebra over a real number field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-12T22:39:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "betti numbers",
      "general type",
      "fold product",
      "upper half plane",
      "real number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.3384D"
    }
  }
}