Mathematische Annalen

, Volume 340, Issue 1, pp 209–222

Generic vanishing and minimal cohomology classes on abelian varieties



We establish a—and conjecture further—relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing of the cohomology of twisted ideal sheaves. The main ingredient is the Generic Vanishing criterion established in Pareschi G. and Popa M. (GV-sheaves, Fourier–Mukai transform, and Generic Vanishing. Preprint math.AG/0608127), based on the Fourier–Mukai transform.

Mathematics Subject Classification (2000)

14K12 14F17 


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  1. 1.
    Beauville A. (1982). Sous-variétés spéciales des variétés de Prym. Compositio Math. 45: 357–383 MATHMathSciNetGoogle Scholar
  2. 2.
    Beauville, A.: Quelques remarques sur la transformation the Fourier dans l’anneau de Chow d’une variété abélienne. Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982, Springer LNM, vol. 1016, pp. 238–260 (1983)Google Scholar
  3. 3.
    Clemens H. and Griffiths P. (1972). The intermediate Jacobian of the cubic threefold. Ann. Math. 95: 281–356 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Debarre O. (1995). Minimal cohomology classes and Jacobians. J. Algebr. Geom. 4: 321–335 MATHMathSciNetGoogle Scholar
  5. 5.
    Debarre O. (1995). Fulton–Hansen and Barth–Lefschetz theorems for subvarieties of abelian varieties. J. Reine Angew. Math. 467: 187–197 MATHMathSciNetGoogle Scholar
  6. 6.
    Eisenbud D. and Goto S. (1984). Linear freee resolutions and minimal multiplicity. J. Algebra 88: 89–133 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fulton W. (1998). Intersection theory, 2nd edn. Springer, Heidelberg Google Scholar
  8. 8.
    Grothendieck, A., Dieudonné, J.: Eléments de Géométrie Algébrique, III, Étude cohomologique des faisceaux coherents. Publ. Math. IHES 11 (1961) and 17 (1963)Google Scholar
  9. 9.
    Green M. and Lazarsfeld R. (1987). Deformation theory, generic vanishing theorems and some conjectures of Enriques, Catanese and Beauville. Invent. Math. 90: 389–407 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hacon Ch. (2004). A derived category approach to generic vanishing. J. Reine Angew. Math. 575: 173–187 MATHMathSciNetGoogle Scholar
  11. 11.
    Höring, A.: M-regularity of the Fano surface. Preprint arXiv:0704.0558Google Scholar
  12. 12.
    Hoyt William L. (1963). On products and algebraic families of Jacobian varieties. Ann. Math. 77: 415–423 CrossRefMathSciNetGoogle Scholar
  13. 13.
    Lange H. and Birkenhake Ch. (2004). Complex abelian varieties, 2nd edn. Springer, Heidelberg Google Scholar
  14. 14.
    Mukai S. (1981). Duality between D(X) and \(D(\widehat{X})\) with its application to Picard sheaves. Nagoya Math. J. 81: 153–175 MATHMathSciNetGoogle Scholar
  15. 15.
    Mukai, S.: Fourier functor and its application to the moduli of bundles on an abelian variety. In: Algebraic Geometry, Sendai 1985, Advanced studies in pure mathematics, vol. 10, pp. 515–550 (1987)Google Scholar
  16. 16.
    Okonek, Ch., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Progr. Math., vol.3. Birkhäuser (1980)Google Scholar
  17. 17.
    Pareschi G. and Popa M. (2003). Regularity on abelian varieties I. J. Am. Math. Soc. 16: 285–302 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pareschi, G., Popa, M.: Castelnuovo theory and the geometric Schottky problem. J. Reine Angew. Math. Preprint math.AG/0407370 (to appear)Google Scholar
  19. 19.
    Pareschi, G., Popa, M.: GV-sheaves, Fourier–Mukai transform, and Generic Vanishing. Preprint math.AG/0608127Google Scholar
  20. 20.
    Ran Z. (1981). On subvarieties of abelian varieties. Invent. Math. 62: 459–479 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Dipartamento di MatematicaUniversità di RomaRomaItaly
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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