ar X iv : m at h / 99 06 12 9 v 2 [ m at h . A G ] 1 4 D ec 1 99 9 ALGEBRAIC GAUSS - MANIN SYSTEMS AND BRIESKORN MODULES

We study the algebraic Gauss-Manin system and the algebraic Brieskorn module associated to a polynomial mapping with isolated singularities. Since the algebraic GaussManin 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 GaussManin 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 twodimensional case we can describe the global structure of the algebraic Gauss-Manin system rather explicitly.