MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $GL(r, C)$?

share|cite|improve this question
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)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.