{
  "id": "math/0610881",
  "version": "v1",
  "published": "2006-10-28T21:58:21.000Z",
  "updated": "2006-10-28T21:58:21.000Z",
  "title": "On the $D$-dimension of a certain type of threefolds",
  "authors": [
    "Jing Zhang"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, \\Omega^j_Y)=0$ for all $j\\geq 0$, $i>0$ and $h^0(Y, {\\mathcal{O}}_Y) > 1$. Let $X$ be a smooth completion of $Y$ such that the boundary $X-Y$ is the support of an effective divisor $D$ on $X$ with simple normal crossings. We prove that the $D$-dimension of $X$ cannot be 2, i.e., either any two nonconstant regular functions are algebraically dependent or there are three algebraically independent nonconstant regular functions on $Y$. Secondly, if the $D$-dimension of $X$ is greater than 1, then the associated scheme of $Y$ is isomorphic to Spec$\\Gamma(Y, {\\mathcal{O}}_Y)$. Furthermore, we prove that an algebraic manifold $Y$ of any dimension $d\\geq 1$ is affine if and only if $H^i(Y, \\Omega^j_Y)=0$ for all $j\\geq 0$, $i>0$ and it is regularly separable, i.e., for any two distinct points $y_1$, $y_2$ on $Y$, there is a regular function $f$ on $Y$ such that $f(y_1)\\neq f(y_2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-28T21:58:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J30",
      "32Q28"
    ],
    "keywords": [
      "algebraically independent nonconstant regular functions",
      "algebraic manifold",
      "threefolds",
      "simple normal crossings",
      "smooth completion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10881Z"
    }
  }
}