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Strongly semistable sheaves and the Mordell–Lang conjecture over function fields

  • Damian RösslerEmail author
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Abstract

We give a new proof of the Mordell–Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer’s theorem that the Harder–Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. The interest of this proof is that it provides simple effective bounds (depending on the degree of the canonical line bundle) for the degree of the isotrivial finite cover whose existence is predicted by the Mordell–Lang conjecture. We also present a conjecture on the Harder–Narasimhan filtration of the cotangent bundle of a smooth projective variety of general type in positive characteristic and a conjectural refinement of the Bombieri–Lang conjecture in positive characteristic.

Notes

References

  1. 1.
    Abramovich, D.: Subvarieties of semiabelian varieties. Compos. Math. 90(1), 37–52 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Benoist, F., Bouscaren, E., Pillay, A.: On function field Mordell–Lang and Manin–Mumford. J. Math. Log. 16(1), 1650001, 24 (2016).  https://doi.org/10.1142/S021906131650001X
  3. 3.
    Dirigé par M. Demazure et A. Grothendieck: Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux. In: Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Lecture Notes in Mathematics, vol. 152, Springer, Berlin (1962/1964)Google Scholar
  4. 4.
    Gillet, H., Rössler, D.: Rational points of varieties with ample cotangent bundle over function fields. Math. Ann. 371(3–4), 1137–1162 (2018).  https://doi.org/10.1007/s00208-017-1569-4 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hrushovski, E.: The Mordell–Lang conjecture for function fields. J. Am. Math. Soc. 9(3), 667–690 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics, E31. Friedrich Vieweg & Sohn, Braunschweig (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Langer, A.: Semistable sheaves in positive characteristic. Ann. Math. (2) 159(1), 251–276 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Martin-Deschamps, M.: Propriétés de descente des variétés à fibré cotangent ample. Ann. Inst. Fourier (Grenoble) 34(3), 39–64 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8, 2nd edn. Cambridge University Press, Cambridge (1989). Translated from the Japanese by M. ReidzbMATHGoogle Scholar
  10. 10.
    Rössler, D.: On the Manin–Mumford and Mordell–Lang conjectures in positive characteristic. Algebra Number Theory 7(8), 2039–2057 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Samuel, P.: Compléments à un article de Hans Grauert sur la conjecture de Mordell. Inst. Hautes Études Sci. Publ. Math. 29, 55–62 (1966)CrossRefzbMATHGoogle Scholar
  12. 12.
    Shepherd-Barron, N.I.: Semi-stability and reduction mod p. Topology 37(3), 659–664 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Séminaire sur les Pinceaux de Courbes de Genre au Moins Deux, Astérisque, vol. 86, Société Mathématique de France, Paris (1981)Google Scholar
  14. 14.
    Yau, S.T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. USA 74(5), 1798–1799 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ziegler, P.: Mordell–Lang in positive characteristic. Rend. Semin. Mat. Univ. Padova 134, 93–131 (2015).  https://doi.org/10.4171/RSMUP/134-3 MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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