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Comments on Sampson’s approach toward Hodge conjecture on Abelian varieties

  • Tuyen Trung Truong
Article

Abstract

Let A be an Abelian variety of dimension n. For \(0<p<2n\) an odd integer, Sampson constructed a surjective homomorphism \(\pi {:}\,J^p(A)\rightarrow A\), where \(J^p(A)\) is the higher Weil Jacobian variety of A. Let \({\widehat{\omega }}\) be a fixed form in \(H^{1,1}(J^p(A),{\mathbb {Q}})\), and \(N=\dim (J^p(A))\). He observes that if the map \(\pi _*({\widehat{\omega }}^{N-p-1}\wedge .){:}\, H^{1,1}(J^p(A),{\mathbb {Q}})\rightarrow H^{n-p,n-p}(A,{\mathbb {Q}})\) is injective, then the Hodge conjecture is true for A in bidegree (pp). In this paper, we give some clarification of the approach and show that the map above is not injective except some special cases where the Hodge conjecture is already known. We propose a modified approach.

Keywords

Abelian varieties Hodge conjecture Weil’s Jacobian varieties 

Mathematics Subject Classification

14Kxx 14Cxx 32xxx 

References

  1. 1.
    Lange, H., Birkenhake, C.: Complex Abelian Varieties. Grund. der math. Wiss., vol. 302. Springer, Berlin (1992)Google Scholar
  2. 2.
    Lewis, J.D.: A Survey of the Hodge Conjecture, CRM Monograph Series, vol. 10, 2nd edn. American Mathematical Society, Providence. Appendix 2 there in is by B. B. Gordon (1999)Google Scholar
  3. 3.
    Sampson, J.H.: Higher Jacobians and cycles on Abelian varieties. Compos. Math. 47(2), 133–147 (1982)MathSciNetzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of MathematicsKorea Institute for Advanced StudySeoulRepublic of Korea
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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