{
  "id": "math/0605351",
  "version": "v4",
  "published": "2006-05-13T15:42:51.000Z",
  "updated": "2007-11-05T13:38:07.000Z",
  "title": "A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces",
  "authors": [
    "Alexei Skorobogatov",
    "Yuri Zarhin"
  ],
  "comment": "20 pages Final version; to appear in the Journal of Algebraic Geometry",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $k$ be a field that is finitely generated over the field of rational numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely irreducible smooth projective variety over $k$, let $Br(X)$ be the cohomological Brauer-Grothendieck group of $X$ and $Br_0(X)$ the image of $Br(k)$ in $Br(X)$. We write $Br_1(X)$ for the subgroup of elements in $Br(X)$ that become trivial after replacing $k$ by its algebraic closure. We prove that $Br(X)/Br_0(X)$ is finite if $X$ is a $K3$ surface. When $X$ is (a principal homogeneous space of) an abelian variety over $k$ then we prove that $Br(X)/Br_1(X)$ is finite.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2007-11-05T13:38:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G35",
      "14G25"
    ],
    "keywords": [
      "abelian variety",
      "brauer group",
      "finiteness theorem",
      "k3 surfaces",
      "irreducible smooth projective variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......5351S"
    }
  }
}