Advertisement

manuscripta mathematica

, Volume 152, Issue 1–2, pp 61–125 | Cite as

Around the Thom–Sebastiani theorem, with an appendix by Weizhe Zheng

  • Luc IllusieEmail author
Article

Abstract

For germs of holomorphic functions \(f: (\mathbf {C}^{m+1},0) \rightarrow (\mathbf {C},0)\), \(g: (\mathbf {C}^{n+1},0) \rightarrow (\mathbf {C},0)\) having an isolated critical point at 0 with value 0, the classical Thom–Sebastiani theorem describes the vanishing cycles group \(\Phi ^{m+n+1}(f \oplus g)\) (and its monodromy) as a tensor product \(\Phi ^m(f) \otimes \Phi ^n(g)\), where \((f \oplus g)(x,y) = f(x) + g(y), x = (x_0,{\ldots },x_m), y = (y_0,{\ldots },y_n)\). We prove algebraic variants and generalizations of this result in étale cohomology over fields of any characteristic, where the tensor product is replaced by a certain local convolution product, as suggested by Deligne. They generalize Fu (Math Res Lett 21:101–119, 2014). The main ingredient is a Künneth formula for \(R\Psi \) in the framework of Deligne’s theory of nearby cycles over general bases. In the last section, we study the tame case, and the relations between tensor and convolution products, in both global and local situations.

Mathematics Subject Classification

Primary: 14F20 Secondary: 11T23 18F10 32S30 32S40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Deligne, P.: Les constantes des équations fonctionnelles des fonctions \(L\). Séminaire à l’IHÉS, (1980)Google Scholar
  2. 2.
    Deligne, P.: Letter to N. A’ Campo, Nov. 14, (1972)Google Scholar
  3. 3.
    Deligne, P.: Letter to L. Fu. April 13 (2011)Google Scholar
  4. 4.
    Deligne, P.: Letter to L. Illusie. May 5 (1999)Google Scholar
  5. 5.
    Deligne, P.: La conjecture de Weil : II. Pub. Math. IHÉS 52, 137–252 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ekedahl, T.: On the Adic Formalism. The Grothendieck Festschrift, Vol. II, 197–218. Birkhäuser, Basel (1990)Google Scholar
  7. 7.
    Fu, L.: A Thom-Sebastiani theorem in characteristic p. Math. Res. Lett. 21, 101–119 (2014). arXiv:1105.5210 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Illusie, L., Laszlo, Y., Orgogozo, F. (eds.): Gabber’s work on local uniformization and étale cohomology of quasi-excellent schemes. Seminar at École Polytechnique 2006–2008. (Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l’École Polytechnique 2006–2008.) (French, English) Zbl 1297.14003 Astérisque 363–364. Paris: Société Mathématique de France (SMF) (2014)Google Scholar
  9. 9.
    Illusie, L.: I Autour du théorème de monodromie locale. In: Périodes \(p\)-adiques, Séminaire de Bures 1988, Astérisque 223, 9-57, SMF (1994)Google Scholar
  10. 10.
    Illusie, L.: On semistable reduction and the calculation of nearby cycles. In: Adolphson, A., Baldassarri, F., Berthelot, P., Katz, N., Loeser, F. (eds.) Geometric Aspects of Dwork Theory. Vol. I, II, pp. 785–803. Walter de Gruyter, Berlin (2004)Google Scholar
  11. 11.
    Illusie, L.: Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne), in Caractéristique d’Euler-Poincaré, 161-172. Astérisque 82, Soc. Math. de France (1981)Google Scholar
  12. 12.
    Illusie, L.: XI Produits orientés, in [8]Google Scholar
  13. 13.
    Illusie, L.: Perversité et variation. Manuscr. math. 112, 271–295 (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kashiwara, M., Schapira, P.: Categories and sheaves, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 332, Springer, Berlin (2006)Google Scholar
  15. 15.
    Katz, N.: Gauss Sums, Kloosterman Sums, and Monodromy Groups, Ann. of Math. Studies, vol. 116. Princeton Univ. Press (1988)Google Scholar
  16. 16.
    Katz, N.M.: Local-to-global extensions of representations of fundamental groups. Ann. Inst. Fourier (Grenoble) 36(4), 69–106 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Laumon, G.: Caractéristique d’Euler-Poincaré des faisceaux constructibles sur une surface. Analysis and topology on singular spaces. II, III (Luminy, 1981), volume 101 of Astérisque, pp. 193–207. Soc. Math. France, Paris (1983)Google Scholar
  18. 18.
    Laumon, G.: Comparaison de caractéristiques d’Euler-Poincaré en cohomologie \(\ell \)-adique. Comptes Rendus de l’Acad. des Sc., Série I. Mathématique, 292, no. 3 (1981), 209-12Google Scholar
  19. 19.
    Laumon, G.: Majoration de sommes exponentielles attachées aux hypersurfaces diagonales. Ann. scient. Éc. Norm. Sup., 4e série, t. 16, 1–58 (1983)Google Scholar
  20. 20.
    Laumon, G.: Vanishing cycles over a base of dimension \(\ge 1\). In Algebraic geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes in Math., pp. 143–150. Springer, Berlin (1983)Google Scholar
  21. 21.
    Laumon, G.: Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil. Pub. Math. IHÉS 65, 131–210 (1987)CrossRefzbMATHGoogle Scholar
  22. 22.
    Orgogozo, F.: II Topologies adaptées à l’uniformisation locale, in [8]Google Scholar
  23. 23.
    Orgogozo, F.: Modifications et cycles proches sur une base générale, Int. Math. Res. Not., posted on 2006, Art. ID 25315, 38, doi: 10.1155/IMRN/2006/25315 (French)
  24. 24.
    Orgogozo, F.: XII-A Descente cohomologique orientée, in [8]Google Scholar
  25. 25.
    Rojas-León, A.: Local convolution of \(\ell \)-adic sheaves on the torus. Math. Z. 274(3–4), 1211–1230 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Saito, T.: Characteristic cycle of the exterior product of constructible sheaves (2016), arXiv:1607.03157v1
  27. 27.
    Saito, T.: The characteristic cycle and the singular support of a constructible sheaf Invent. Math. (2016, to appear). arXiv:1510.03018v4
  28. 28.
    Saito, T.: Wild ramification and the cotangent bundle. JAG (to appear). arXiv:1301.4632v6
  29. 29.
    Saito, T.: \(\epsilon \)-factor of a tamely ramified sheaf on a variety. Invent. Math. 113(2), 389–417 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sawin, W.: Local Convolution on Elliptic Curves and \({{\bf G}}_m\), in preparationGoogle Scholar
  31. 31.
    Sebastiani, M., Thom, R.: Un résultat sur la monodromie. Invent. Math. 13, 90–96 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    SGA 4 1/2, Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie, par P. Deligne, Lecture Notes in Math. 569, Springer (1977)Google Scholar
  33. 33.
    SGA 4 Théorie des topos et cohomologie étale des schémas, Séminaire de géométrie algébrique du Bois-Marie 1963-64, dirigé par M. Artin, A. Grothendieck, J.-L. Verdier, Lecture Notes in Math. 269, 270, 305, Springer (1972, 1973)Google Scholar
  34. 34.
    SGA 5, Cohomologie \(\ell \)-adique et fonctions \(\rm L\), Séminaire de géométrie algébrique du Bois-Marie 1965-66, dirigé par A. Grothendieck, Lecture Notes in Math. 589, Springer (1977)Google Scholar
  35. 35.
    SGA 7, Groupes de monodromie en géométrie algébrique, Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969, I dirigé par A. Grothendieck, II par P. Deligne et N. Katz, Lecture Notes in Math. 288, 340, Springer (1972, 1973)Google Scholar

Copyright information

© © European Union 2016

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance

Personalised recommendations