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Logarithmic Riemann–Hilbert correspondences for rigid varieties

Diao, Hansheng and Lan, Kai-Wen and Liu, Ruochuan and Zhu, Xinwen (2022) Logarithmic Riemann–Hilbert correspondences for rigid varieties. Journal of the American Mathematical Society . ISSN 0894-0347. doi:10.1090/jams/1002. (In Press) https://resolver.caltech.edu/CaltechAUTHORS:20220707-978151000

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Abstract

On any smooth algebraic variety over a p-adic local field, we construct a tensor functor from the category of de Rham p-adic étale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a p-adic analogue of Deligne's classical Riemann--Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this p-adic Riemann--Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1090/jams/1002DOIArticle
https://arxiv.org/abs/1803.05786arXivDiscussion Paper
ORCID:
AuthorORCID
Lan, Kai-Wen0000-0003-0795-220X
Zhu, Xinwen0000-0003-1614-9216
Additional Information:© 2022 American Mathematical Society. Received by editor(s): August 24, 2018. Received by editor(s) in revised form: December 22, 2019, July 25, 2021, August 13, 2021, August 14, 2021, January 27, 2022, and February 15, 2022. Published electronically: June 24, 2022. The second author was partially supported by the National Science Foundation under agreement No. DMS-1352216, by an Alfred P. Sloan Research Fellowship, and by a Simons Fellowship in Mathematics. The third author was partially supported by the National Natural Science Foundation of China under agreement Nos. NSFC-11571017 and NSFC-11725101, and by the Tencent Foundation. The fourth author was partially supported by the National Science Foundation under agreement Nos. DMS-1602092 and DMS-1902239, by an Alfred P. Sloan Research Fellowship, and by a Simons Fellowship in Mathematics. Any opinions, findings, and conclusions or recommendations expressed in this writing are those of the authors, and do not necessarily reflect the views of the funding organizations. We would like to thank Kiran Kedlaya, Koji Shimizu, and Daxin Xu for helpful conversations, and thank the Beijing International Center for Mathematical Research, the Morningside Center of Mathematics, and the California Institute of Technology for their hospitality. Some important ideas occurred to us when we were participants of the activities at the Mathematical Sciences Research Institute and the Oberwolfach Research Institute for Mathematics, and we would like to thank these institutions for providing stimulating working environments. Finally, we would like to thank Yihang Zhu and the anonymous referees for many helpful comments that helped us correct and improve earlier versions of this paper.
Funders:
Funding AgencyGrant Number
NSFDMS-1352216
Alfred P. Sloan FoundationUNSPECIFIED
Simons FoundationUNSPECIFIED
National Natural Science Foundation of China11571017
National Natural Science Foundation of China11725101
Tencent FoundationUNSPECIFIED
NSFDMS-1602092
NSFDMS-1902239
Classification Code:2020 Mathematics Subject Classification. Primary 14F40, 14G22; Secondary 14D07, 14F30, 14G35.
DOI:10.1090/jams/1002
Record Number:CaltechAUTHORS:20220707-978151000
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20220707-978151000
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:115419
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:12 Jul 2022 14:58
Last Modified:12 Jul 2022 14:58

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