Relative cohomology with respect to a Lefschetz pencil
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Hossein Movasati
Abstract
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.
© Walter de Gruyter
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- Traces of CM values of modular functions
- Représentations semi-stables de torsion dans le cas er < p − 1
- Zero cycles on a threefold with isolated singularities
- On the density of algebraic foliations without algebraic invariant sets
- Compactness of solutions to some geometric fourth-order equations
- Relative cohomology with respect to a Lefschetz pencil
- There are genus one curves of every index over every number field
- The uniqueness of Cuntz-Krieger type algebras
Articles in the same Issue
- Traces of CM values of modular functions
- Représentations semi-stables de torsion dans le cas er < p − 1
- Zero cycles on a threefold with isolated singularities
- On the density of algebraic foliations without algebraic invariant sets
- Compactness of solutions to some geometric fourth-order equations
- Relative cohomology with respect to a Lefschetz pencil
- There are genus one curves of every index over every number field
- The uniqueness of Cuntz-Krieger type algebras
