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Transactions of the American Mathematical Society

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Characteristic class and the $\varepsilon$-factor of an étale sheaf


Authors: Naoya Umezaki, Enlin Yang and Yigeng Zhao
Journal: Trans. Amer. Math. Soc. 373 (2020), 6887-6927
MSC (2010): Primary 14F20; Secondary 11G25, 11S40
DOI: https://doi.org/10.1090/tran/8187
Published electronically: August 6, 2020
MathSciNet review: 4155195
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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.


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Additional Information

Naoya Umezaki
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
MR Author ID: 1171997
Email: umezaki@ms.u-tokyo.ac.jp, umezakinaoya@gmail.com

Enlin Yang
Affiliation: School of Mathematical Sciences, Peking University, No.5 Yiheyuan Road Haidian District., Beijing, 100871, People’s Republic of China
Email: yangenlin@math.pku.edu.cn, enlin.yang@mathematik.uni-regensburg.de

Yigeng Zhao
Affiliation: School of Sciences, Westlake University, 310024 Hangzhou, People’s Republic of China; and Institute of Sciences, Westlake Institute for Advanced Study, 310024 Hangzhou, People’s Republic of China
MR Author ID: 1278775
Email: zhaoyigeng@westlake.edu.cn, yigeng.zhao@mathematik.uni-regensburg.de

Received by editor(s): November 7, 2018
Received by editor(s) in revised form: June 14, 2019, and September 5, 2019
Published electronically: August 6, 2020
Additional Notes: Enlin Yang is the corresponding author.
The second author was partially supported by NSFC grants 11901008, and Alexander von Humboldt Foundation for his research at Universität Regensburg and Freie Universität Berlin.
Both the second and the third authors were partially supported by the DFG through CRC 1085 Higher Invariants (Universität Regensburg). The authors are grateful to these institutions.
Article copyright: © Copyright 2020 American Mathematical Society