Let $\rho: \pi\_1(S,s\_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V\_{s\_0} = V$.


Let $H$ be the Zariski closure of the image of $\rho$ in $GL(V)$. We say that $\rho$ is big if the identity component $H^0$ of $H$ acts irreducibly on $V$.


Clearly, if $\rho$ is big, then $H$ acts irreducibly on $V$.


There should be a vast amount of monodromy representations coming from geometry (e.g., coverings of Riemann surfaces, families of abelian varieties, Lefschetz pencils) so that $H$ acts irreducibly on $V$, but $H^0$ doesn't. I'm looking for explicit and natural examples.


**Question.** Which monodromy representations are irreducible, but not big?



What are the easiest examples of irreducible, but not big, monodromy representations
I have seen "big monodromy" used before, in some papers of Katz I think, with a somewhat different meaning (basically that $H^0$ should as big as possible). But I'll use your definition, since that's what you seem to be interested in. One can certainly construct many non big irreducible representations as follows: Start with a variety $X$ on which a finite group $G$ in such a way that $V= H^i(X,\mathbb{Q})$ is irreducible as a $\mathbb{Q}[G]$ or $\overline{\mathbb{Q}}[G]$-module (you didn't specify which).
Let $Y$ be a variety on which $G$ acts freely, then $(X\times Y)/G\to Y/G$ is big in your sense, although isotrivial. For a specific example, let $G=S\_3$ acting on a Fermat cubic in $\mathbb{P}^2$, and $Y=\mathbb{A}^3$ minus the big diagonal.


