Characteristic cycle of a rank one sheaf and ramification theory

Yatagawa, Yuri

doi:10.1090/jag/758

Author: Yuri Yatagawa
Journal: J. Algebraic Geom. 29 (2020), 471-545
DOI: https://doi.org/10.1090/jag/758
Published electronically: March 9, 2020
MathSciNet review: 4158459
Full-text PDF

Abstract | References | Additional Information

Abstract: We compute the characteristic cycle of a rank one sheaf on a smooth surface over a perfect field of positive characteristic. We construct a canonical lifting on the cotangent bundle of Kato’s logarithmic characteristic cycle using ramification theory and prove the equality of the characteristic cycle and the canonical lifting. As corollaries, we obtain a computation of the singular support in terms of ramification theory and the Milnor formula for the canonical lifting.

References

Ahmed Abbes and Takeshi Saito, Ramification of local fields with imperfect residue fields, Amer. J. Math. 124 (2002), no. 5, 879–920. MR 1925338
Ahmed Abbes and Takeshi Saito, The characteristic class and ramification of an $l$-adic étale sheaf, Invent. Math. 168 (2007), no. 3, 567–612. MR 2299562, DOI https://doi.org/10.1007/s00222-007-0040-7
Ahmed Abbes and Takeshi Saito, Analyse micro-locale $l$-adique en caractéristique $p>0$: le cas d’un trait, Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 25–74 (French, with English and French summaries). MR 2512777, DOI https://doi.org/10.2977/prims/1234361154
Ahmed Abbes and Takeshi Saito, Ramification and cleanliness, Tohoku Math. J. (2) 63 (2011), no. 4, 775–853. MR 2872965, DOI https://doi.org/10.2748/tmj/1325886290
A. Beilinson, Constructible sheaves are holonomic, Selecta Math. (N.S.) 22 (2016), no. 4, 1797–1819. MR 3573946, DOI https://doi.org/10.1007/s00029-016-0260-z
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
Spencer Bloch, Cycles on arithmetic schemes and Euler characteristics of curves, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 421–450. MR 927991
Jean-Luc Brylinski, Théorie du corps de classes de Kato et revêtements abéliens de surfaces, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 23–38 (French, with English summary). MR 723946
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
Kazuya Kato, Swan conductors for characters of degree one in the imperfect residue field case, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 101–131. MR 991978, DOI https://doi.org/10.1090/conm/083/991978
Kazuya Kato, Class field theory, ${\scr D}$-modules, and ramification on higher-dimensional schemes. I, Amer. J. Math. 116 (1994), no. 4, 757–784. MR 1287939, DOI https://doi.org/10.2307/2375001
Shigeki Matsuda, On the Swan conductor in positive characteristic, Amer. J. Math. 119 (1997), no. 4, 705–739. MR 1465067
Arthur Ogus, Lectures on logarithmic algebraic geometry, Cambridge Studies in Advanced Mathematics, vol. 178, Cambridge University Press, Cambridge, 2018. MR 3838359
Takeshi Saito, Wild ramification and the characteristic cycle of an $l$-adic sheaf, J. Inst. Math. Jussieu 8 (2009), no. 4, 769–829. MR 2540880, DOI https://doi.org/10.1017/S1474748008000364
Takeshi Saito, Wild ramification of schemes and sheaves, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 335–356. MR 2827799
Takeshi Saito, Wild ramification and the cotangent bundle, J. Algebraic Geom. 26 (2017), no. 3, 399–473. MR 3647790, DOI https://doi.org/10.1090/S1056-3911-2016-00681-1
Takeshi Saito, The characteristic cycle and the singular support of a constructible sheaf, Invent. Math. 207 (2017), no. 2, 597–695. MR 3595935, DOI https://doi.org/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 https://doi.org/10.24033/asens.2339
Takahiro Tsushima, On localizations of the characteristic classes of $\ell $-adic sheaves of rank 1, Algebraic number theory and related topics 2007, RIMS Kôkyûroku Bessatsu, B12, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, pp. 193–207. MR 2605781
Yuri Yatagawa, Equality of two non-logarithmic ramification filtrations of abelianized Galois group in positive characteristic, Doc. Math. 22 (2017), 917–952. MR 3665400, DOI https://doi.org/10.1177/1081286515616049

Additional Information

Yuri Yatagawa
Affiliation: Department of Mathematics, Saitama University, Saitama, 338-8570, Japan
MR Author ID: 1218497
Email: yatagawa@mail.saitama-u.ac.jp, yatagawa.math@gmail.com

Received by editor(s): January 11, 2018
Received by editor(s) in revised form: August 1, 2019, and November 22, 2019
Published electronically: March 9, 2020
Additional Notes: This work was supported in part by the Program for Leading Graduate Schools, MEXT, Japan, JSPS KAKENHI Grant Number 15J03851, and the CRC 1085 Higher Invariants at the University of Regensburg.
Article copyright: © Copyright 2020 University Press, Inc.