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


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.

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  1. 1.

    Abramovich, D.: Subvarieties of semiabelian varieties. Compos. Math. 90(1), 37–52 (1994)

    MathSciNet  MATH  Google 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).

    MathSciNet  Article  Google Scholar 

  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)

  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).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Hrushovski, E.: The Mordell–Lang conjecture for function fields. J. Am. Math. Soc. 9(3), 667–690 (1996)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics, E31. Friedrich Vieweg & Sohn, Braunschweig (1997)

    Google Scholar 

  7. 7.

    Langer, A.: Semistable sheaves in positive characteristic. Ann. Math. (2) 159(1), 251–276 (2004)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Martin-Deschamps, M.: Propriétés de descente des variétés à fibré cotangent ample. Ann. Inst. Fourier (Grenoble) 34(3), 39–64 (1984)

    MathSciNet  Article  Google 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. Reid

    Google Scholar 

  10. 10.

    Rössler, D.: On the Manin–Mumford and Mordell–Lang conjectures in positive characteristic. Algebra Number Theory 7(8), 2039–2057 (2013)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  12. 12.

    Shepherd-Barron, N.I.: Semi-stability and reduction mod p. Topology 37(3), 659–664 (1998)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Séminaire sur les Pinceaux de Courbes de Genre au Moins Deux, Astérisque, vol. 86, Société Mathématique de France, Paris (1981)

  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)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ziegler, P.: Mordell–Lang in positive characteristic. Rend. Semin. Mat. Univ. Padova 134, 93–131 (2015).

    MathSciNet  Article  MATH  Google Scholar 

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Rössler, D. Strongly semistable sheaves and the Mordell–Lang conjecture over function fields. Math. Z. 294, 1035–1049 (2020).

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