Selecta Mathematica

, Volume 24, Issue 5, pp 4279–4292 | Cite as

Cohomologically rigid local systems and integrality

  • Hélène EsnaultEmail author
  • Michael Groechenig


We prove that the monodromy of an irreducible cohomologically complex rigid local system with finite determinant and quasi-unipotent local monodromies at infinity on a smooth quasiprojective complex variety X is integral. This answers positively a special case of a conjecture by Carlos Simpson. On a smooth projective variety, the argument relies on Drinfeld’s theorem on the existence of \(\ell \)-adic companions over a finite field. When the variety is quasiprojective, one has in addition to control the weights and the monodromy at infinity.

Mathematics Subject Classification

14D07 14G15 14F35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abe, T., Esnault, H.: A Lefschetz theorem for overconvergent isocrystals with Frobenius structure (2016). Accessed 19 Mar 2018
  2. 2.
    Abramovich, D., Corti, A., Vistoli, A.: Twisted bundles and admissible covers. Commun. Algebra 31(8), 3547–3618 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bass, H.: Groups of integral representation type. Pacific J. Math. 86(1), 15–51 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, 5–171 (1982)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Deligne, P.: Théorème d’intégralité, Appendix to “Le niveau de la cohomologie des intersections complètes” by N. Katz, Exposé XXI in SGA 7, Lecture Notes in Math. vol. 340, pp. 363–400. Springer, New York (1973)Google Scholar
  6. 6.
    Deligne, P.: Les constantes des équations fonctionnelles des fonctions \(L\), Proc. Antwerpen Conference, vol. 2, Lecture Notes in Mathematics, vol. 349, pp. 501–597. Springer, BerlinGoogle Scholar
  7. 7.
    Deligne, P.: La conjecture de Weil: II. Publ. math. de l’I.H.É.S. 52, 137–252 (1980)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Deligne, P.: Le groupe fondamental de la droite projective moins trois points, in “Galois groups over \({\mathbb{Q}}\)”, MSRI Publications, vol. 16, pp. 72–297. Springer (1989)Google Scholar
  9. 9.
    Drinfeld, V.: On a conjecture of Deligne. Moscow Math. J. 12(3), 515–542 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Esnault, H., Groechenig, M.: Rigid connections, \(F\)-isocrystals and integrality (2017). Accessed 19 Mar 2018
  11. 11.
    Grothendieck, A.: Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois-Marie, Lecture Notes in Mathematics, vol. 224. Springer, Berlin (1971)Google Scholar
  12. 12.
    Katz, N.: Local-to-global extensions of representations of fundamental groups. Annales de l’Institut Fourier 36(4), 69–106 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kerz, M., Schmidt, A.: On different notions of tameness in arithmetic geometry. Math. Ann. 346(3), 641–668 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lafforgue, L.: Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 346(3), 641–668 (2010)MathSciNetGoogle Scholar
  15. 15.
    Langer, A., Simpson, C.: Rank 3 rigid representations of projective fundamental groups (2016).
  16. 16.
    Saito, T.: Wild ramification and the cotangent bundle. J. Algebraic Geom. 26, 399–473 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Simpson, C.: Higgs bundles and local systems. Publ. math. de l’I.H.É.S. 75, 5–95 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Freie Universität BerlinBerlinGermany

Personalised recommendations