Abstract
In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito’s logarithmic ramification filtration. This provides a positive answer to the main conjecture in [24] for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11901287
Funding source: Natural Science Foundation of Jiangsu Province
Award Identifier / Grant number: BK20190288
Funding statement: The first author is currently supported by the National Natural Science Foundation of China (No. 11901287), the Natural Science Foundation of Jiangsu Province (No. BK20190288) and the Nanjing Science and Technology Innovation Project.
Acknowledgements
We would like to express our gratitude to A. Abbes, A. Beilinson, L. Fu, O. Gabber, F. Orgogozo, T. Saito and Y. Tian for inspiring discussions on this topic and valuable suggestions. We are indebted to T. Saito for providing a simplified argument in the proof of Theorem 4.3, as well as for suggesting Proposition 7.7, which gave birth to the last section of this paper. We thank the anonymous referee for careful reading and valuable comments. The first author would like to thank Y. Cao and T. Deng for helpful discussions on abelian varieties. This manuscript was mainly written while the first author was visiting the Max-Planck Institute for Mathematics in Bonn and while the second author was visiting the Catholic University in Leuven and received support from the long term structural funding-Methusalem grant of the Flemish Government. We would like to thank both Institutes for their hospitality and for providing outstanding working conditions.
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