{
  "id": "math/0312239",
  "version": "v1",
  "published": "2003-12-11T14:56:01.000Z",
  "updated": "2003-12-11T14:56:01.000Z",
  "title": "Threefolds with Vanishing Hodge Cohomology",
  "authors": [
    "Jing Zhang"
  ],
  "comment": "24 pages, accepted by Transactions of AMS",
  "categories": [
    "math.AG"
  ],
  "abstract": "We consider algebraic manifolds $Y$ of dimension 3 over $\\Bbb{C}$ with $H^i(Y, \\Omega^j_Y)=0$ for all $j\\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the $D$-dimension of $X$ is not zero, then $Y$ is a fibre space over a smooth affine curve $C$ (i.e., we have a surjective morphism from $Y$ to $C$ such that general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of $X$ is $-\\infty$ and the $D$-dimension of X is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of $Y$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-12-11T14:56:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J30",
      "14B15",
      "14C20"
    ],
    "keywords": [
      "vanishing hodge cohomology",
      "threefolds",
      "smooth affine curve",
      "higher direct images",
      "smooth completion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....12239Z"
    }
  }
}