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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 $GL(r, C)$?

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up vote 7 down vote accepted

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)

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