Logarithmic Riemann–Hilbert correspondences for rigid varieties

Abstract

On any smooth algebraic variety over a p p -adic local field, we construct a tensor functor from the category of de Rham p 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 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 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.