LOGARITHMIC RIEMANN–HILBERT CORRESPONDENCES FOR RIGID VARIETIES

Hansheng Diao, Kai Wen Lan, Ruochuan Liu, Xinwen Zhu

Research output: Contribution to journalArticlepeer-review

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Abstract

On any smooth algebraic variety over a padic local field, we construct a tensor functor from the category of de Rham padic étale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a padic 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 padic 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.

Original languageEnglish (US)
Pages (from-to)483-562
Number of pages80
JournalJournal of the American Mathematical Society
Volume36
Issue number2
DOIs
StatePublished - 2023

Bibliographical note

Funding Information:
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.

Funding Information:
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.

Publisher Copyright:
© 2023, American Mathematical Society.

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