TITLE: Chow motive of a fibration
QUESTION [7 upvotes]: If we have a fibration of smooth projective complex varieties $F\to E\to B$, which is locally trivial in the analytic topology, and the global monodromy is trivial. Then is it true that the Chow motives satisfy $h(E)\cong h(F)\otimes h(B)$? I am pretty sure this should fail, but cannot think of a counterexample.

REPLY [4 votes]: I know very little about motives, but conjecturally you should expect that $h(E) \cong h(F) \otimes h(B)$. 
Your hypotheses imply that $E$ and $F \times B$ have isomorphic cohomology in the category of $\mathbf Q$-Hodge structures. In fact if $E \to B$ is smooth and proper, and $B$ is smooth and connected, then Saito's theory of mixed Hodge modules implies degeneration of the Leray spectral sequence and that the Leray filtration splits in the category of mixed Hodge structures. The additional assumptions of locally trivial (I'm assuming this means in the category of complex analytic spaces) and trivial monodromy imply also that $H^p(B,R^q f_\ast \mathbf Q) \cong H^p(B) \otimes H^q(F)$. (It also implies more directly that Leray spectral sequence degenerates.)
In any case, the conjectural functor (pure/mixed motives) $\to$ (pure/mixed $\mathbf Q$-Hodge structures) is conjectured to be fully faithful. This statement contains in particular the Hodge conjecture. So you should expect $E$ and $F\times B$ to have isomorphic motives, although proving this may be hard.
On a sidenote, do you have any nontrivial examples of such fibrations? I have a hard time imagining one.