Semi-continuity of conductors, and ramification bound of nearby cycles

Haoyu Hu

doi:10.1515/crelle-2023-0060

Article

Semi-continuity of conductors, and ramification bound of nearby cycles

Published/Copyright: October 4, 2023

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2023 Issue 804

Abstract

For a constructible étale sheaf on a smooth variety of positive characteristic ramified along an effective divisor, the largest slope in Abbes and Saito’s ramification theory of the sheaf gives a divisor with rational coefficients called the conductor divisor. In this article, we prove decreasing properties of the conductor divisor after pull-backs. The main ingredient behind is the construction of étale sheaves with pure ramifications. As applications, we first prove a lower semi-continuity property for conductors of étale sheaves on relative curves in the equal characteristic case, which supplement Deligne and Laumon’s lower semi-continuity property of Swan conductors ([33]) and is also an ℓ -adic analogue of André’s semi-continuity result of Poincaré–Katz ranks for meromorphic connections on complex relative curves. ([6]). Secondly, we give a ramification bound for the nearby cycle complex of an étale sheaf ramified along the special fiber of a regular scheme semi-stable over an equal characteristic henselian trait, which extends a main result in a joint work with Teyssier ([20]) and answers a conjecture of Leal ([35]) in a geometric situation.

Funding statement: This research is partially supported by the National Natural Science Foundation of China (Grants No. 11901287 and No. 11971223), the Natural Science Foundation of Jiangsu Province (Grant No. BK20190288), the Fundamental Research Funds for the Central Universities and the Nanjing Science and Technology Innovation Project.

Acknowledgements

The author would like to express his gratitude to T. Saito for inspiring comments and valuable suggestions. Particularly, his comments improve results in Section 5 and Section 7. The author thanks J.-B. Teyssier for helpful discussions on applications and on related topic in D-modules theory. The author thanks an anonymous referee for useful comments and remarks.

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Received: 2021-07-20

Revised: 2023-07-17

Published Online: 2023-10-04

Published in Print: 2023-11-01

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https://doi.org/10.1515/crelle-2023-0060