Relative Cohomology with Respect to a Lefschetz Pencil

Let M be a complex projective manifold of dimension n+1 and f a meromorphic function on M obtained by a generic pencil of hyperplane sections of M . The n-th cohomology vector bundle of f0 = f |M−R, where R is the set of indeterminacy points of f , is defined on the set of regular values of f0 and we have the usual Gauss-Manin connection on it. Following Brieskorn’s methods in [Br], we extend the n-th cohomology vector bundle of f0 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 f0. 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 f 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 f0.