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In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A general setup for such a result in algebraic geometry is:

Given a proper, generically smooth map $\pi:X \rightarrow S$ of relative dimension d, say S is connected. This gives rise to an $l$-adic representations of the etale fundamental group $\pi_1(U)$ where $U$ is smooth locus of $\pi$ corresponding to higher pushforward $R^d \pi_* Q_l$. One might say it has "big monodromy" if the Zariski closure of the image is as big as it can be given that it has to respect cup-product, etc.

My specific question is what are the geometric consequences of big monodromy? If we know such a result for $\pi$, what does that say about the geometry of the fibration or at the very least is there geometric intuition for what it should mean?

I welcome intuition from number theory, algebraic geometry, or complex geometry.

I have also heard that "one should expect big monodromy unless there is a reason not to" (for example, complex multiplication). What are other examples of things which inhibit big monodromy?

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I found this question a little vague, but let me at least remark on "other examples of things which inhibit big monodromy." Mumford gives an example in section 4 of

D. Mumford, “A note of Shimura’s paper “Discontinuous groups and abelian varieties”,” Math. Ann. 181 (1969), 345–351.

of an abelian variety A whose Galois representation has image strictly smaller than Sp_{2g}(Z_p), despite the fact that End(A) = Z. The keyword to look up is "Mumford-Tate group", which is in some sense the answer to the question

How big COULD the Galois representation on an abelian variety be, subject to all geometric 'things which inhibit big monodromy'?

Reference comes from a paper of Chris Hall which shows how to prove big monodromy results in many cases.

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