Characteristic class and the 𝜖-factor of an étale sheaf

Umezaki, Naoya; Yang, Enlin; Zhao, Yigeng

doi:10.1090/tran/8187

Characteristic class and the $\varepsilon$-factor of an étale sheaf
HTML articles powered by AMS MathViewer

by Naoya Umezaki, Enlin Yang and Yigeng Zhao PDF

Trans. Amer. Math. Soc. 373 (2020), 6887-6927 Request permission

Abstract:

We prove a twist formula for the $\varepsilon$-factor of a constructible sheaf on a projective smooth variety over a finite field in terms of characteristic class of the sheaf. This formula is a modified version of the formula conjectured by Kato and Saito in [Ann. of Math. 168 (2008), pp. 33–96].

We give two applications of the twist formula. First, we prove that the characteristic classes of constructible étale sheaves on projective smooth varieties over a finite field are compatible with proper push-forward. Secondly, we show that the two Swan classes in the literature are the same on proper smooth surfaces over a finite field.

References

Tomoyuki Abe and Deepam Patel, On a localization formula of epsilon factors via microlocal geometry, Ann. K-Theory 3 (2018), no. 3, 461–490. MR 3830199, DOI 10.2140/akt.2018.3.461
Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652
Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
A. Beilinson, Constructible sheaves are holonomic, Selecta Math. (N.S.) 22 (2016), no. 4, 1797–1819. MR 3573946, DOI 10.1007/s00029-016-0260-z
A. Beilinson, Topological $\scr E$-factors, Pure Appl. Math. Q. 3 (2007), no. 1, Special Issue: In honor of Robert D. MacPherson., 357–391. MR 2330165, DOI 10.4310/PAMQ.2007.v3.n1.a13
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
P. Deligne and G. Henniart, Sur la variation, par torsion, des constantes locales d’équations fonctionnelles de fonctions $L$, Invent. Math. 64 (1981), no. 1, 89–118 (French). MR 621771, DOI 10.1007/BF01393935
Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520
P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Séminaire à l’ I.H.E.S., 1980, notes de L. Illusie.
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 501–597 (French). MR 0349635
Lei Fu, Etale cohomology theory, Revised edition, Nankai Tracts in Mathematics, vol. 14, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. MR 3380806, DOI 10.1142/9569
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), no. 2, 327–402. MR 833194, DOI 10.1007/BF01388811
A. Grothendieck et al., Cohomologie $\ell$-adique et fonctions L, Séminaire de Géométrie Algébrique du Bois-Marie 1965–1966 (SGA 5). dirigé par A. Grothendieck avec la collaboration de I. Bucur, C. Houzel, L. Illusie, J.-P. Jouanolou et J-P. Serre. Lecture Notes in Mathematics 589, Springer-Verlag, Berlin-Heidelberg-New York, (1977).
A. Grothendieck, Récoltes et Semailles, Réflexions et témoignages sur un passé de mathématicien, http://lipn.univ-paris13.fr/\~{}duchamp/Books\&more/Grothendieck/RS/pdf/RetS.pdf
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
L. Illusie, Y. Lazslo, and F. Orgogozo, Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents, Astérisque, vol. 361-362, Soc. Math. France, 2014, Séminaire à l’École Polytechnique 2006-2008.
Luc Illusie, Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne), The Euler-Poincaré characteristic (French), Astérisque, vol. 82, Soc. Math. France, Paris, 1981, pp. 161–172 (French). MR 629127
Kazuya Kato and Shuji Saito, Unramified class field theory of arithmetical surfaces, Ann. of Math. (2) 118 (1983), no. 2, 241–275. MR 717824, DOI 10.2307/2007029
Kazuya Kato and Takeshi Saito, Ramification theory for varieties over a perfect field, Ann. of Math. (2) 168 (2008), no. 1, 33–96. MR 2415398, DOI 10.4007/annals.2008.168.33
Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052, DOI 10.1515/9781400882120
Nicholas M. Katz, Travaux de Laumon, Astérisque 161-162 (1988), Exp. No. 691, 4, 105–132 (1989). Séminaire Bourbaki, Vol. 1987/88. MR 992205
Reinhardt Kiehl and Rainer Weissauer, Weil conjectures, perverse sheaves and $l$’adic Fourier transform, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 42, Springer-Verlag, Berlin, 2001. MR 1855066, DOI 10.1007/978-3-662-04576-3
G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210 (French). MR 908218
Masayoshi Nagata, A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ. 3 (1963), 89–102. MR 158892, DOI 10.1215/kjm/1250524859
Deepam Patel, de Rham ${\scr E}$-factors, Invent. Math. 190 (2012), no. 2, 299–355. MR 2981817, DOI 10.1007/s00222-012-0381-8
D. Patel, K-theory of algebraic microdifferential operators, preprint.
Wayne Raskind, Abelian class field theory of arithmetic schemes, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 85–187. MR 1327282
Shuji Saito, Functional equations of $L$-functions of varieties over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 2, 287–296. MR 763423
T. Saito, Characteristic cycles and the conductor of direct image, arXiv:1704.04832.
Takeshi Saito, On the proper push-forward of the characteristic cycle of a constructible sheaf, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 485–494. MR 3821182
Takeshi Saito, The characteristic cycle and the singular support of a constructible sheaf, Invent. Math. 207 (2017), no. 2, 597–695. MR 3595935, DOI 10.1007/s00222-016-0675-3
Takeshi Saito and Yuri Yatagawa, Wild ramification determines the characteristic cycle, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 4, 1065–1079 (English, with English and French summaries). MR 3679621, DOI 10.24033/asens.2339
Takeshi Saito, $\epsilon$-factor of a tamely ramified sheaf on a variety, Invent. Math. 113 (1993), no. 2, 389–417. MR 1228131, DOI 10.1007/BF01244312
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
Jean-Pierre Serre, Groupes de Grothendieck des schémas en groupes réductifs déployés, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 37–52 (French). MR 231831
J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607
J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305–347. MR 0217026
N. Umezaki, E. Yang, and Y. Zhao, A blow up formula for Gysin pull-back, https://yangenlin.files.wordpress.com/2018/08/buf.pdf.
Isabelle Vidal, Formule du conducteur pour un caractère $l$-adique, Compos. Math. 145 (2009), no. 3, 687–717 (French, with English and French summaries). MR 2507745, DOI 10.1112/S0010437X08003850
Isabelle Vidal, Formule de torsion pour le facteur epsilon d’un caractère sur une surface, Manuscripta Math. 130 (2009), no. 1, 21–44 (French, with English and French summaries). MR 2533765, DOI 10.1007/s00229-009-0267-2
Weizhe Zheng, Six operations and Lefschetz-Verdier formula for Deligne-Mumford stacks, Sci. China Math. 58 (2015), no. 3, 565–632. MR 3319927, DOI 10.1007/s11425-015-4970-z