If $f: X \to S$ is a projective smooth morphism between complex algebraic varieties. Does the $\pi\_1(S)$-representation corresponding to the local system $R^i f\_\* (C\_X)$ on $S$ maps $\pi\_1(S)$ onto a discrete subgroup of $GL(r, C)$?



Is the image of the representation of the fundamental group associated to a local system discrete?
Yes, because it lies in $GL(r, \mathbb{Z})$ (use universal coefficients: $R^if\_\*(\mathbb{C}\_X)= R^if\_\*(\mathbb{Z}\_X)\otimes \mathbb{C}$). A more interesting question -- which was open for a while -- was whether the monodromy group is always arithmetic. The answer turned out to be no. See Nori "A nonarithmetic monodromy group" Compte Rendus (1986)


