A note on the Gauss-Manin connection for abelian schemes

2022/01/17 by Tiago Fonseca, Fonseca, Tiago J., Nils Matthes +1
Mathematics · Physics and Astronomy · #14F40 (Primary) 14K05 #32M25 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2201.06402

openalex publication_date 2022/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study differential forms on the universal vector extension A^\natural of an abelian scheme A in characteristic zero, and derive a new construction of the D-group scheme structure on A^\natural. This gives, in particular, a rather simple description of the Gauss--Manin connection on the de Rham cohomology of A in terms of global algebraic differential forms on A^\natural. The key ingredient is the computation of the coherent cohomology of A^\natural, due to Coleman and Laumon.

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