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Algebraic Gauss-Manin systems and Brieskorn modules
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 123, Number 1, February 2001
- pp. 163-184
- 10.1353/ajm.2001.0001
- Article
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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 nonquasi-coherent sheaf which gives the cohomology of every fiber. Then we study the algebraic Brieskorn module, and show that its position in 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.