2020/02/29 by Phùng Hô Hái, Phùng Hô Hai, João Pedro dos Santos
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Differential (mechanical device) #Discrete mathematics #Group (periodic table) #Integrable system #Logarithm #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Physics #Pure mathematics #Ring (chemistry) #Scheme (mathematics) #Spectrum (functional analysis) #Variety (cybernetics) #math.AG #msc:14F10 #msc:14L15 #msc:32C15 #msc:35Q15
paper · pdf · doi:10.2422/2036-2145.202006_010
31 pages, final version, to appear in The Annali della Scuola Normale Superiore di Pisa
openalex publication_date 2022/02/14 · arxiv created 2022/03/05 · arxiv updated 2022/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Deligne's celebrated "Riemann--Hilbert correspondence" relates representations of the fundamental group of a smooth complex algebraic variety and regular-singular integrable connections. In this work, we show how to arrive at a similar statement in the case of a smooth scheme X over the spectrum of a ring R=\mathbb C[[t1,…, tr]]/I. On one side of the correspondence we have representations on R-modules of the fundamental group of the special fibre, and on the other we have certain integrable R-connections admitting logarithmic models. The correspondence is then applied to give explicit examples of differential Galois groups of \mathbb C[[t]]--connections.