Survey on some aspects of Lefschetz theorems in algebraic geometry

Abstract

We survey classical material around Lefschetz theorems for fundamental groups, and show the relation to parts of Deligne’s program in Weil II.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Abe, T., Esnault, H.: A Lefschetz theorem for overconvergent isocrystals with Frobenius structure. http://www.mi.fu-berlin.de/users/esnault/preprints/helene/123_abe_esn (2016)

  2. 2.

    Abe, T.: Langlands correspondence for isocrystals and existence of crystalline companions for curves. arXiv:1310.0528v1

  3. 3.

    Bhatt, B., Schloze, P.: The pro-étale topology for schemes. Astérisque 369, 99–201 (2015)

    MATH  Google Scholar 

  4. 4.

    Bott, R.: On a theorem of Lefschetz. Michigan Math. J 6, 211–216 (1959)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Crew, R.: F-isocrystals and p-adic representations, Algebraic Geometry, Bowdoin (1985). In: Proceedings of Symposia in Pure Mathematics 46, Part 2, pp 111–138. American Mathematical Society, Providence (1987)

  6. 6.

    Crew, R.: Specialization of crystalline cohomology. Duke Math. J. 53(3), 749–757 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Deligne, P.: La conjecture de Weil II. Publ. Math. Inst. Hautes Études Sci. 52, 137–252 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Deligne, P.: Finitude de l’extension de \({\mathbb{Q}}\) engendrée par des traces de Frobenius, en caractéristique finie. Mosc. Math. J 12(3), 497–514 (2012). (668)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Drinfeld, V.: On a conjecture of Deligne. Mosc. Math. J. 12(3), 515–542 (2012). (668)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Esnault, H. : A remark on Deligne’s finiteness theorem, Int. Math. Res. Not. http://www.mi.fu-berlin.de/users/esnault/preprints/helene/121_finiteness (2016, in print)

  11. 11.

    Esnault, H., Kerz, M.: A finiteness theorem for Galois representations of function fields over finite fields (after Deligne). Acta Math. Vietnam. 37(4), 531–562 (2012)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Esnault, H., Kindler, L.: Lefschetz theorems for tamely ramified coverings. Proc. Am. Math. Soc. 144, 5071–5080 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Forster, O.: Lectures on Riemann Surfaces. Graduate Text in Mathematics, vol. 81. Springer, New York (1981)

    Google Scholar 

  14. 14.

    Fundamental groups of schemes. http://stacks.math.columbia.edu

  15. 15.

    Grothendieck, A.: Caractérisation et classification des groupes à type multiplicatif, SGA3 Exp. X

  16. 16.

    Grothendieck, A.: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, Séminaire de Géométrie Algébrique du Bois-Marie, Adv. St.in Pure Math., vol. 2. North-Holland Publ. Co. (1962)

  17. 17.

    Grothendieck, A.: Éléments de Géométrie Algébrique IV, Études locales des schémas et des morphismes de schémas, Publ. math. I. H. É. S., vol. 62 (1967)

  18. 18.

    Grothendieck, A.: Éléments de Géométrie Algébrique III, Études cohomologiques des faisceaux cohérents, Publ. math. I. H. É. S., vol. 11 (1961)

  19. 19.

    Grothendieck, A.: Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois-Marie, Lecture Notes in Mathematics, vol. 224 (1971)

  20. 20.

    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  21. 21.

    Kedlaya, K.: Notes on Isocrystals. http://kskedlaya.org/papers/isocrystals (2016, preprint)

  22. 22.

    Kedlaya, K.: Semistable reduction for overconvergent F-isocrystals, I: unipotence and logarithmic extensions. Compos. Math. 143, 1164–1212 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Kerz, M., Schmidt, A.: On different notions of tameness in arithmetic geometry. Math. Ann. 346(3), 641–668 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Koshikawa, T.: Overconvergent unit-root \(F\)-isocrystals are isotrivial. arXiv: 1511.02884v2

  25. 25.

    Lafforgue, L.: Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147(1), 1–241 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Lefschetz, S.: L’Analysis situs et la géométrie algébrique. Gauthier-Villars, Paris (1950)

  27. 27.

    Milne, J.S.: Fields and Galois Theory. http://www.jmilne.org/math/CourseNotes/ft.html

  28. 28.

    Poincaré, H.: Analysis situs. J. École Polytech. 1(2), 1–123 (1895)

    MATH  Google Scholar 

  29. 29.

    Riemann, B.: Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Inauguraldissertation, Göttingen (1851)

  30. 30.

    Riemann, B.: Theorie der Abel’schen functionen. J. Reine Angew. Math. 54, 101–155 (1857)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Scholze, P.: p-adic Hodge theory for rigid analytic varieties. In: Forum of Mathematics, Pi, vol. 1 (2013)

  32. 32.

    Serre, J.-P.: Corps locaux, Publ. de l’Univ. de Nancago, VIII, Hermann, Paris (1962)

  33. 33.

    Serre, J.-P.: Exemples de variétés projectives conjuguées non homéomorphes. Ann. Inst. Fourier 6, 20–50 (1955)

    Google Scholar 

  34. 34.

    Szamuely, T.: Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics, vol. 117, p. 270 (2009)

  35. 35.

    Tsuzuki, N.: Morphism of \(F\)-isocrystals and the finite monodromy theorem for unit-root \(F\)-isocrystals. Duke Math. J. 111, 385–419 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Wiesend, G.: A construction of covers of arithmetic schemes. J. Number. Theory 121(1), 131–181 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Wiesend, G.: Tamely ramified covers of varieties and arithmetic schemes. Forum Math. 20(3), 515–522 (2008)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

It is a pleasure to thank Jakob Stix for a discussion on separable base points reflected in Sect. 3. We thank Tomoyuki Abe and Atsushi Shiho for discussions. We thank the public of the Santaló lectures at the Universidad Complutense de Madrid (October 2015) and the Rademacher lectures at the University of Pennsylvania (February 2016), where some points discussed in those notes were presented. In particular, we thank Ching-Li Chai for an enlightening discussion on compatible systems. We thank Moritz Kerz for discussions we had when we tried to understand Deligne’s program in Weil II while writing [11]. We thank the two referees for their friendly and thorough reports which helped us to improve the initial version of these notes.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hélène Esnault.

Additional information

Supported by the Einstein program.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Esnault, H. Survey on some aspects of Lefschetz theorems in algebraic geometry. Rev Mat Complut 30, 217–232 (2017). https://doi.org/10.1007/s13163-017-0223-8

Download citation

Keywords

  • Fundamental group
  • Lefschetz theorems
  • Lisse sheaves
  • Isocrystals

Mathematics Subject Classification

  • 14F20
  • 14F45
  • 14G17
  • 14G99