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Let $X$ be a smooth, projective algebraic variety over a field of characteristic zero. Let $U \subset X$ be an open subvariety such that $D=X \setminus U$ is a normal crossing divisor. Let $\mathcal{E}$ be a local system on $U$. I would like to know how to compute the cohomology

$H^k(U, \mathcal{E})$

by means of Lefschetz pencils. So let us us choice a Lefschetz pencil on $X$ whose base locus intersects properly the components of $D$. After blowing up the base locus, one gets a morphism $\rho: \tilde{X} \to \mathbb{P}^1$.

(1) Can anybody help me to use Leray spectral sequence to relate this cohomology group with the cohomology of (an open of) $\mathbb{P}^1$ with values in $R^k\rho_\ast \mathcal{E}$?

(2) Suppose we know the cohomology of the fibers of $\rho$ with values in $\mathcal{E}$. What can one say about $H^k(U, \mathcal{E})$?

Thanks a lot

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