1999/06/18 by Alexandru Dimca, Dimca, Alexandru, Morihiko Saito +1
Mathematics · Physics and Astronomy · #32S40 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons #math.AG #msc:32S40
paper · pdf · doi:10.48550/arxiv.math/9906129
AMSTeX, 20 pages, revised shorter version of RIMS-1218 with title changed
openalex publication_date 1999/06/18 · arxiv created 1999/12/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the algebraic Gauss-Manin system and the algebraic Brieskorn module associated to a polynomial mapping with isolated singularities. Since the algebraic Gauss-Manin system does not contain any information on the cohomology of singular fibers, we first construct a non quasi-coherent sheaf which gives the cohomology of every fiber. Then we study the algebraic Brieskorn module, and show that its position in the the algebraic Gauss-Manin system is determined by a natural map to quotients of local analytic Gauss-Manin systems, and its pole part by the vanishing cycles at infinity, comparing it with the Deligne extension. This implies for example a formula for the determinant of periods. In the two-dimensional case we can describe the global structure of the algebraic Gauss-Manin system rather explicitly.