Abstract.
A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. Their proof uses analytic methods. We refine and give a completely new proof of their result. Our proof combines galois-theoretic methods and algebraic geometry in positive characteristic. When the variety has a model over a function field and its Picard variety has no isotrivial factors, we show how to replace the galois-theoretic results we need by results from model theory (mathematical logic). Furthermore, we prove partial analogs of the conjecture of Beauville and Catanese in positive characteristic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21 Springer-Verlag, Berlin, 1990
Bouscaren, E.: Proof of the Mordell-Lang conjecture for function fields. In: Model theory and algebraic geometry. An introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture. Edited by Elisabeth Bouscaren. Lecture Notes in Mathematics, 1696. Springer-Verlag, Berlin, 1998
Deligne, P., Illusie, L.: Relèvements modulo p2 et décomposition du complexe de de Rham. Invent. Math. 89 (2), 247–270 (1987)
Lang, S.: Fundamentals of Diophantine geometry. Springer-Verlag, New York, 1983
Green, M., Lazarsfeld, R.: Higher obstructions to deforming cohomology groups of line bundles. J. Amer. Math. Soc. 4 (1), 87–103 (1991)
Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. Publ. Math. Inst. Hautes Etud. Sci.
Hrushovski, E.: Proof of Manin’s theorem by reduction to positive characteristic. In: Model theory and algebraic geometry. An introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture. Edited by Elisabeth Bouscaren. Lecture Notes in Mathematics, 1696. Springer-Verlag, Berlin, 1998
Nori, M.V.: On subgroups of
Invent. Math. 88, 257–275 (1987)Pink, R.: On the Order of the Reduction of a Point on an Abelian Variety. Math. Ann. (2004) DOI: 10.1007/s00208-004-0548-8
Pink, R., Roessler, D.: On psi-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture. To appear in the Journal of Algebraic Geometry
Simpson, C.: Subspaces of moduli spaces of rank one local systems. Ann. Sci. École Norm. Sup. (4) 26 (3), 361–401 (1993)
Wong, S.: Power residues on abelian varieties. Manuscripta Math. 102 (1), 129–138 (2000)
Zarhin, Y.: A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction. Invent. Math. 79, 309–321 (1985)
Invent. Math. 88, 257–275 (1987)Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pink, R., Roessler, D. A conjecture of Beauville and Catanese revisited. Math. Ann. 330, 293–308 (2004). https://doi.org/10.1007/s00208-004-0549-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-004-0549-7