Logarithmic Riemann-Hilbert correspondences for rigid varieties

Hansheng Diao Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02141 Kai-Wen Lan School of Mathematics, University of Minnesota Twin Cities, Minneapolis MN 55455 Ruochuan Liu International Center for Mathematical Research (BICMR), Beijing (Peking) University, Beijing 100871, PR China Xinwen Zhu Department of Mathematics, California Institute of Technology, Pasadena CA 91125

Algebraic Geometry mathscidoc:2203.45008

On any smooth algebraic variety over a p-adic local field, we construct a tensor functor from the category of de Rham p-adic ́etale 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.
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  title={Logarithmic Riemann-Hilbert correspondences for rigid varieties},
  author={Hansheng Diao, Kai-Wen Lan, Ruochuan Liu, and Xinwen Zhu},
Hansheng Diao, Kai-Wen Lan, Ruochuan Liu, and Xinwen Zhu. Logarithmic Riemann-Hilbert correspondences for rigid varieties. 2019. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317164816272104001.
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