Relative Cohomology with Respect to a Lefschetz Pencil

TL;DR

This paper extends the cohomology bundle and Gauss-Manin connection for a meromorphic function on a complex projective manifold, establishing a link with Brieskorn modules and analyzing their meromorphic sections.

Contribution

It generalizes the cohomology bundle and Gauss-Manin connection to include meromorphic extensions using differential forms, connecting to Brieskorn modules in a new setting.

Findings

The extended connection is meromorphic at critical values.

Meromorphic sections with poles relate to the Brieskorn module.

The Brieskorn module is a free polynomial module of rank equal to the fiber's cohomology dimension.

Abstract

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 $f_0=f|_{M-\RR}$, where $\RR$ is the set of indeterminacy points of $f$, is defined on the set of regular values of $f_0$ and we have the usual Gauss-Manin connection on it. Following Brieskorn's methods in [bri], we extend the $n$-th cohomology vector bundle of $f_0$ and the associated Gauss-Manin connection to $\pl$ by means of differential forms. The new connection turns out to be meromorphic on the critical values of $f_0$. We prove that the meromorphic global sections of the vector bundle with poles of arbitrary order at $\infty\in\pl$ is isomorphic to the Brieskorn module of $f$ in a natural way, and so the Brieskorn module in this case is a free $\Pf$-module of rank $\beta_n$, where $\Pf$ is the ring of polynomials in $t$ and $\beta_n$ is the dimension of $n$-th cohomology group of a regular fiber of $f_0$.