variation of (polarized) Hodge structure

Question

Suppose $\mathbb{V}$ is the underlining local system of a variation of (polarized) rational Hodge structure over base $B$. This local system corresponds to the monodromy representation $$\pi_1(B,b)\rightarrow GL(\mathbb{V}_b).$$

If the monodromy representation is irreducible, then the local system $\mathbb{V}$ does not have any non-trivial local subsystem and vice versa.

My question is the following.

1. Suppose we only know $\mathbb{V}$ does not admit any nontrivial sub variation of (polarized) Hodge structure. What can we say about the irreducibiity of the underlining local system? Is it still irreducible or there are counterexamples?

2. Under the same assumption on $\mathbb{V}$ as above, is it true $\mathbb{V}_b$ is an irreducible Hodge structure for $b\in B$ very general?

Any comments/input are appreciated.

Answer 1

No to one of your questions; here is an easy counterexample. Pick any base $B$ and let $A$ be a simple complex abelian variety then the pullback of $H^1(A,\mathbb{Q})$ to $B$ is a trivial, and therefore reducible, local system. But it admits no nontrivial sub variations of Hodge structure.

Added The newly added question is more interesting. Yes, if $V$ is a VHS which is irreducible as a local system, then $V_b$ is irreducible as a Hodge structure for very general $b$. This follows for example from prop 7.5 of Deligne, La conjecture de Weil pour les surfaces K3. It's probably also in André's paper that jmc referenced [it's lemma 4 in André, I just checked].