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

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• 作者：Luc Illusie
• 关键词：mixed Hodge theory ; $${\ell}$$ ?adic sheaf ; ramification ; Swan conductor ; Euler–Poincaré characteristic ; characteristic cycle ; logarithmic structure ; 32S35 ; 14F20 ; 11S15 ; 14C20
• 刊名：Japanese Journal of Mathematics
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：10
• 期：2
• 页码：237-248
• 全文大小：211 KB
• 参考文献：1.Abbes A., Saito T.: Ramification of local fields with imperfect residue fields. Amer. J. Math., 124, 879-20 (2001)MathSciNet CrossRef
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  3.A. Abbes and T. Saito, The characteristic class and ramification of an \({\ell}\) -adic étale sheaf, Invent. Math., 168 (2007), 567-12.
  4.A. Abbes and T. Saito, Ramification and cleanliness, Tohoku Math. J. (2), Centennial Issue, 63 (2011), 775-53.
  5.Beilinson A.: p-adic periods and de Rham cohomology. J. Amer. Math. Soc., 25, 715-38 (2012)MathSciNet CrossRef MATH
  6.A. Beilinson, Constructible sheaves are holonomic, preprint, arXiv:-505.-6768 .
  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. 421-50.
  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. 3-34, 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, 573-76 (1981)MathSciNet MATH
  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), 247-70.
  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, 41-1 (1981)MathSciNet MATH
  15.G. Faltings, F-isocrystals on open varieties: results and conjectures, In: The Grothendieck Festschrift. Vol. II, Progr. Math., 87, Birkh?user Boston, MA, 1990, pp. 219-48.
  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 1965-6, 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, 1978-979, Astérisque, 82-83, Soc. Math. France, Paris, 1981, pp. 161-72.
  19.L. Illusie and W. Zheng, Odds and ends on finite group actions and traces, Int. Math. Res. Not. IMRN, 2013, 1-2; Errata and addenda, Int. Math. Res. Not. IMRN, 2014, 2572-576.
  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. 315-42.
  21.K. Kato, Logarithmic structures of Fontaine–Illusie, In: Algebraic Analysis, Geometry, and Number Theory, Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191-24.
  22.K. Kato, Class field theory, \({\mathcal{D}}\) -modules, and ramification on higher-dimensional schemes. I, Amer. J. Math., 116 (1994), 757-84.
  23.Kato K., Saito T.: On the conductor formula of Bloch. Publ. Math. Inst. Hautes études Sci., 100, 5-51 (2004)MathSciNet CrossRef MATH
  24.Kato K., Saito T.: Ramification theory for varieties over a perfect field. Ann. of Math. (2) 168, 33-6 (2008)MathSciNet CrossRef MATH
  25.Kato K., Saito T.: Ramification theory for varieties over a local field. Publ. Math. Inst. Hautes études Sci., 117, 1-78 (2013)MathSciNet CrossRef MATH
  26.G. Laumon, Semi-continuité du conducteur de Swan (d’après P. Deligne), In: Caractéristique d’Euler–Poincaré, Séminaire ENS, 1978-979, Astérisque, 82-83, Soc. Math. France, Paris, 1981, pp. 173-19.
  27.G. Laumon, Comparaison de caractéristiques d’Euler–Poincaré en cohomologie \({\ell}\) -adique, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 209-12.
  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. 193-07.
  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, 423-32 (1974)MathSciNet CrossRef MATH
  31.Olsson M.C.: Logarithmic geometry and algebraic stacks. Ann. Sci. école Norm. Sup. (4) 36, 747-91 (2003)MathSciNet MATH
  32.Olsson M.C.: The logarithmic cotangent complex. Math. Ann., 333, 859-31 (2005)MathSciNet
• 作者单位：Luc Illusie (1)
  
  1. Département de Mathématiques, Batiment 425, Faculté des Sciences d’Orsay, Université Paris Sud 11, F-91405, Orsay Cedex, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  History of Mathematics
  
• 出版者：Springer Japan
• ISSN：1861-3624

文摘

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. Keywords and phrases mixed Hodge theory \({\ell}\)-adic sheaf ramification Swan conductor Euler–Poincaré characteristic characteristic cycle logarithmic structure