Skip to main content
Log in

From Pierre Deligne’s secret garden: looking back at some of his letters

  • Original Article
  • Published:
Japanese Journal of Mathematics Aims and scope

Abstract

I discuss four unpublished letters of Deligne (one on Hodge theory, two on Euler–Poincaré characteristics and ramification of \({\ell}\)-adic sheaves, one on generalized divisors), and sketch some of the developments they generated.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abbes A., Saito T.: Ramification of local fields with imperfect residue fields. Amer. J. Math., 124, 879–920 (2001)

    Article  MathSciNet  Google Scholar 

  2. A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields. II, Doc. Math., Extra Vol.: Kazuya Kato’s Fiftieth Birthday (2003), 5–72.

  3. A. Abbes and T. Saito, The characteristic class and ramification of an \({\ell}\)-adic étale sheaf, Invent. Math., 168 (2007), 567–612.

  4. A. Abbes and T. Saito, Ramification and cleanliness, Tohoku Math. J. (2), Centennial Issue, 63 (2011), 775–853.

  5. Beilinson A.: p-adic periods and de Rham cohomology. J. Amer. Math. Soc., 25, 715–738 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Beilinson, Constructible sheaves are holonomic, preprint, arXiv:1505.06768.

  7. S. Bloch, Cycles on arithmetic schemes and Euler characteristics of curves, In: Algebraic Geometry, Bowdoin, 1985, Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI, 1987, pp. 421–450.

  8. J.-L. Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, In: Géométrie et analyse microlocales, Astérisque, 140-141, Soc. Math. France, Paris, 1986, pp. 3–134, 251.

  9. Brylinski J.-L., Dubson A.S., Kashiwara M.: Formule de l’indice pour modules holonomes et obstruction d’Euler locale. C. R. Acad. Sci. Paris Sér. I Math., 293, 573–576 (1981)

    MathSciNet  MATH  Google Scholar 

  10. P. Deligne, Cohomologie Etale, Séminaire de Géométrie Algébrique du Bois-Marie, Lecture Notes in Math., 569, SGA 4 1/2, Springer-Verlag, 1977.

  11. P. Deligne, Notes sur Euler–Poincaré: brouillon project, handwritten notes, Feb. 8, 2011.

  12. P. Deligne and L. Illusie, Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math., 89 (1987), 247–270.

  13. P. Deligne and N. Katz, Groupes de monodromie en géométrie algébrique, Lecture Notes in Math., 340, SGA 7 II, Springer-Verlag, 1973.

  14. Du Bois P.: Complexe de de Rham filtré d’une variété singulière. Bull. Soc. Math. France, 109, 41–81 (1981)

    MathSciNet  MATH  Google Scholar 

  15. G. Faltings, F-isocrystals on open varieties: results and conjectures, In: The Grothendieck Festschrift. Vol. II, Progr. Math., 87, Birkhäuser Boston, MA, 1990, pp. 219–248.

  16. J.-M. Fontaine, Périodes p-Adiques, Séminaire de Bures, 1988, Astérisque, 223, Soc. Math. France, Paris, 1994.

  17. A. Grothendieck, Cohomologie \({\ell}\)-adique et Fonctions L, Séminaire de Géométrie Algébrique du Bois-Marie 1965–66, Lecture Notes in Math., 589, SGA 5, Springer-Verlag, 1977.

  18. L. Illusie, Théorie de Brauer et caractéristique d’Euler–Poincaré (d’après P. Deligne), In: Caractéristique d’Euler–Poincaré, Séminaire ENS, 1978–1979, Astérisque, 82-83, Soc. Math. France, Paris, 1981, pp. 161–172.

  19. L. Illusie and W. Zheng, Odds and ends on finite group actions and traces, Int. Math. Res. Not. IMRN, 2013, 1–62; Errata and addenda, Int. Math. Res. Not. IMRN, 2014, 2572–2576.

  20. K. Kato, Swan conductors with differential values, In: Galois Representations and Arithmetic Algebraic Geometry, Adv. Stud. Pure Math., 12, North-Holland, Amsterdam, 1987, pp. 315–342.

  21. K. Kato, Logarithmic structures of Fontaine–Illusie, In: Algebraic Analysis, Geometry, and Number Theory, Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224.

  22. K. Kato, Class field theory, \({\mathcal{D}}\)-modules, and ramification on higher-dimensional schemes. I, Amer. J. Math., 116 (1994), 757–784.

  23. Kato K., Saito T.: On the conductor formula of Bloch. Publ. Math. Inst. Hautes études Sci., 100, 5–151 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kato K., Saito T.: Ramification theory for varieties over a perfect field. Ann. of Math. (2) 168, 33–96 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kato K., Saito T.: Ramification theory for varieties over a local field. Publ. Math. Inst. Hautes études Sci., 117, 1–178 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Laumon, Semi-continuité du conducteur de Swan (d’après P. Deligne), In: Caractéristique d’Euler–Poincaré, Séminaire ENS, 1978–1979, Astérisque, 82-83, Soc. Math. France, Paris, 1981, pp. 173–219.

  27. G. Laumon, Comparaison de caractéristiques d’Euler–Poincaré en cohomologie \({\ell}\)-adique, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 209–212.

  28. G. Laumon, Caractéristique d’Euler–Poincaré de faisceaux constructibles sur une surface, In: Analyse et topologie sur les espaces singuliers. II, III, Astérisque, 101-102, Soc. Math. France, Paris, 1983, pp. 193–207.

  29. G. Laumon, Compléments à “Caractéristique d’Euler–Poincaré de faisceaux constructibles sur une surface", thèse, Orsay, 1983.

  30. MacPherson R.D.: Chern classes for singular algebraic varieties. Ann. of Math. (2) 100, 423–432 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  31. Olsson M.C.: Logarithmic geometry and algebraic stacks. Ann. Sci. école Norm. Sup. (4) 36, 747–791 (2003)

    MathSciNet  MATH  Google Scholar 

  32. Olsson M.C.: The logarithmic cotangent complex. Math. Ann., 333, 859–931 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Saito S.: General fixed point formula for an algebraic surface and the theory of Swan representations for two-dimensional local rings. Amer. J. Math., 109, 1009–1042 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  34. T. Saito, Wild ramification and the characteristic cycle of an \({\ell}\)-adic sheaf, J. Inst. Math. Jussieu, 8 (2009), 769–829.

  35. Saito T.: Ramification of local fields with imperfect residue fields III. Math. Ann., 352, 567–580 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Saito, Wild ramification and the cotangent bundle, preprint, arXiv:1301.4632v5.

  37. Saito T.: Characteristic cycle and the Euler number of a constructible sheaf on a surface. J. Math. Sci. Univ. Tokyo, 22, 387–442 (2015)

    MathSciNet  Google Scholar 

  38. T. Saito, The characteristic cycle and the singular support of an étale sheaf, talk at Arithmetic Geometry, Representation Theory and Applications, CIRM, Luminy, June 24, 2015.

  39. Schwede K.: A simple characterization of Du Bois singularities. Compos. Math., 143, 813–828 (2007)

    MathSciNet  MATH  Google Scholar 

  40. Serre J.-P.: Sur la rationalité des représentations d’Artin. Ann. of Math. (2) 72, 405–420 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vidal I.: Théorie de Brauer et conducteur de Swan. J. Algebraic Geom., 13, 349–391 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Vidal I.: Courbes nodales et ramification sauvage virtuelle. Manuscripta Math., 118, 43–70 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Luc Illusie.

Additional information

Communicated by: Takeshi Saito

A talk given on October 5, 2013, at the event Celebrating the Mathematics of Pierre Deligne, IHÉS and the Simons Foundation, New York.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Illusie, L. From Pierre Deligne’s secret garden: looking back at some of his letters. Jpn. J. Math. 10, 237–248 (2015). https://doi.org/10.1007/s11537-015-1514-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11537-015-1514-9

Keywords and phrases

Mathematics Subject Classification (2010)

Navigation