Algebraic Gauss-Manin Systems and Brieskorn Modules

Details Algebraic Gauss-Manin Systems and Brieskorn Modules Algebraic Gauss-Manin Systems and Brieskorn Modules

1999-06-18

arxiv.org/abs/math/9906129v2

Mathematics

Algebraic Geometry

AMSTeX, 20 pages, revised shorter version of RIMS-1218 with title changed

Scientific paper

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.

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