{
  "id": "math/0504142",
  "version": "v3",
  "published": "2005-04-07T14:25:11.000Z",
  "updated": "2005-10-11T20:01:12.000Z",
  "title": "There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology",
  "authors": [
    "Jing Zhang"
  ],
  "comment": "Revised version. Accepted by Michigan Math. J",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "We analyze the structure of the algebraic manifolds $Y$ of dimension 3 with $H^i(Y, \\Omega^j_Y)=0$ for all $j\\geq 0$, $i>0$ and $h^0(Y, {\\mathcal{O}}_Y) > 1$, by showing the deformation invariant of some open surfaces. Secondly, we show when a smooth threefold with nonconstant regular functions satisfies the vanishing Hodge cohomology. As an application, we prove the existence of nonaffine and nonproduct threefolds $Y$ with this property by constructing a family of a certain type of open surfaces parametrized by the affine curve $\\C-\\{0\\}$ such that the corresponding smooth completion $X$ has Kodaira dimension $-\\infty$ and $D$-dimension 1, where $D$ is the effective boundary divisor with support $X-Y$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-10-11T20:01:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14B15",
      "14C20",
      "14J30",
      "32Q28"
    ],
    "keywords": [
      "vanishing hodge cohomology",
      "nontrivial threefolds",
      "open surfaces",
      "nonconstant regular functions satisfies",
      "corresponding smooth completion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4142Z"
    }
  }
}