{
  "id": "math/0610884",
  "version": "v1",
  "published": "2006-10-30T16:22:26.000Z",
  "updated": "2006-10-30T16:22:26.000Z",
  "title": "Hodge Cohomology Criteria For Affine Varieties",
  "authors": [
    "Jing Zhang"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "We give several new criteria for a quasi-projective variety to be affine. In particular, we prove that an algebraic manifold $Y$ with dimension $n$ is affine if and only if $H^i(Y, \\Omega^j_Y)=0$ for all $j\\geq 0$, $i>0$ and $\\kappa(D, X)=n$, i.e., there are $n$ algebraically independent nonconstant regular functions on $Y$, where $X$ is the smooth completion of $Y$, $D$ is the effective boundary divisor with support $X-Y$ and $\\Omega^j_Y$ is the sheaf of regular $j$-forms on $Y$. This proves Mohan Kumar's affineness conjecture for algebraic manifolds and gives a partial answer to J.-P. Serre's Steinness question \\cite{36} in algebraic case since the associated analytic space of an affine variety is Stein [15, Chapter VI, Proposition 3.1].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-30T16:22:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J10",
      "14J30",
      "32E10"
    ],
    "keywords": [
      "hodge cohomology criteria",
      "affine variety",
      "algebraically independent nonconstant regular functions",
      "algebraic manifold",
      "mohan kumars affineness conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10884Z"
    }
  }
}