{
  "id": "1805.01067",
  "version": "v1",
  "published": "2018-05-03T00:42:50.000Z",
  "updated": "2018-05-03T00:42:50.000Z",
  "title": "A remark on the example of Colliot-Thélène and Voisin",
  "authors": [
    "Fumiaki Suzuki"
  ],
  "comment": "6 pages, comments are welcome",
  "categories": [
    "math.AG"
  ],
  "abstract": "A classical question asks whether the Abel-Jacobi map $\\psi^{i} \\colon A^{i}(X)\\rightarrow J^{i}_{a}(X)$ is universal among all regular homomorphisms. Meanwhile, Colliot-Th\\'el\\`ene and Voisin constructed a smooth projective $3$-fold $Y$ such that the Hodge conjecture is false for degree $4$ integral Hodge classes in a strong sense and $H^{i}(Y, \\mathcal{O}_{Y}) = 0$ for all $i>0$. We prove that the Abel-Jacobi map $\\psi^{3}\\colon A^{3}(Y\\times E)\\rightarrow J^{3}_{a}(Y\\times E)$ is not universal for some smooth elliptic curve $E$ if the generalized Bloch conjecture holds for $Y$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-03T00:42:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C25",
      "14C30",
      "14C35"
    ],
    "keywords": [
      "abel-jacobi map",
      "colliot-thélène",
      "integral hodge classes",
      "smooth elliptic curve",
      "generalized bloch conjecture holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}