{
  "id": "math/0405066",
  "version": "v1",
  "published": "2004-05-04T22:06:52.000Z",
  "updated": "2004-05-04T22:06:52.000Z",
  "title": "Convexity of coverings of projective varieties and vanishing theorems",
  "authors": [
    "F. Bogomolov",
    "B. De Oliveira"
  ],
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "This article is concerned with the convexity properties of universal covers of projective varieties. We study the relation between the convexity properties of the universal cover of X and the properties of the pullback map sending vector bundles on X to vector bundles on its universal cover. Our approach motivates a weakened version of the Shafarevich conjecture. We prove this conjecture for projective varieties X whose pullback map identifies a nontrivial extension of a negative vector bundle $V$ by the trivial line bundle with the trivial extension. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holomorphic functions then the pullback map of vector bundles is almost an imbedding. Our methods also give a new proof of the vanishing of the first cohomology for negative vector bundles $V$ over a compact complex manifold $X$ whose rank is smaller than the dimension of X.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-05-04T22:06:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F05",
      "32Q30",
      "32E05",
      "14E20"
    ],
    "keywords": [
      "projective variety",
      "universal cover",
      "vanishing theorems",
      "negative vector bundle",
      "pullback map sending vector bundles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......5066B"
    }
  }
}