Hossein Movasati
Relative cohomology with respect to a Lefschetz pencil
Let M be a complex projective manifold of dimension n + 1 and ƒ a meromorphic function on M obtained by a generic pencil of hyperplane sections of M. The n-th cohomology vector bundle of ƒ0 = ƒ |M−ℛ, where ℛ is the set of indeterminacy points of ƒ, is defined on the set of regular values of ƒ0 and we have the usual Gauss-Manin connection on it. Following Brieskorn's methods in [Brieskorn, E., Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscr. Math. 2 (1970), 103–161.], we extend the n-th cohomology vector bundle of ƒ0 and the associated Gauss-Manin connection to ℙ1 by means of differential forms. The new connection turns out to be meromorphic on the critical values of ƒ0. We prove that the meromorphic global sections of the vector bundle with poles of arbitrary order at ∞ ∈ ℙ1 is isomorphic to the Brieskorn module of ƒ in a natural way, and so the Brieskorn module in this case is a free ℂ[t]-module of rank βn, where ℂ[t] is the ring of polynomials in t and βn is the dimension of n-th cohomology group of a regular fiber of ƒ0.
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2006, 05/2006
Pages: 175 - 199
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