# what does the decomposition theorem say for a Lefschetz pencil?

The setting is the following: let $X$ be a smooth projective variety (say over $\mathbb{C}$), $D$ a simple normal crossings divisor on $X$ and $(H_t)_{t \in \mathbb{P}^1}$ a Lefschetz pencil on $X$ whose axis $\Delta$ intersects transversally all the intersections $D_J:=\bigcap D_{i_1} \cap \cdots \cap D_{i_j}$ (here $J$ is any subset of the set of indexes of the irreducible components of $D$). Consider the blow-up

$\varphi: \tilde{X}:=Bl_{\Delta \cap X} X \to X$

and the morphism induced by the pencil

$\rho: \tilde{X} \to \mathbb{P}^1$.

The assumptions imply that $\varphi^\ast D$ is also a simple normal crossings divisor. Consider a rank one local system $V$ on $X-D$ and its pullback $\tilde{V}$ to $\tilde{X}-\varphi^\ast D$. Look at the intersection complex $IC_{\tilde{X}}(\tilde{V}).$

Now the questions:

1) What does the decomposition theorem say about $R \rho_\ast IC_{\tilde{X}}(\tilde{V})$?

2) Can one determine the different summands in the decomposition?

3) What is the analogue of the fact that $R^j \rho_\ast \mathbb{Q}_{\tilde{X}}$ is constant for $j \neq \dim X-1$ or $\dim X$?

And a more general question:

Does anybody have a good reference where explicit computations involving perverse sheafs and Lefchetz pencils are done?