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Let $X$ be a smooth projective variety of dimension $n$, and let $\pi: C \to \mathbb{P}^1$ be a cover of the projective line, ramified at some points $S=\{p_1, \ldots, p_r\}$. Consider the morphism

$f: X \times C \to \mathbb{P}^1$

obtained by composition of the projection to $C$ with $\pi$.

I am wondering what can be said about the local systems $R^m f_\ast \mathbb{Q}$ on $U=\mathbb{P}^1-S.$

Are they interesting?

Can one compute their monodromy in terms of the monodromy of $\pi$?

What are the cohomologies $H^k_c(U, R^m f_\ast \mathbb{Q})$ for $k=0$ and $1$?

Any help will be appreciated, thanks!

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By an appropriate version of Künneth's formula, $R^mf_\ast\mathbb{Q}= H^m(X,\mathbb{Q})\otimes \pi_\ast\mathbb{Q}$. In particular, the monodromy representation is just a direct sum of copies of the monodromy of $\pi$, which is just the regular representation of the Galois group of $C/\mathbb{P}^1$. Also $H^k_c(U,R^mf_\ast \mathbb{Q})= H^k_c(U)\otimes H^m(X,\mathbb{Q})$.

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