Let $\pi : W \rightarrow Y$ be a holomorphic fibration of complex manifolds. Let $L\rightarrow W$ be a holomorphic line bundle on its total space and denote by 
$$E^k_q := R^q \pi_*L^k$$
the direct image sheaf of its $k$th tensor power on $Y$. 

**Question**: I would like to know if there are interesting examples when $E^k_q$ is a vector bundle for each $q\in \mathbb{N}_0$ and $ k \gg 0$. \
One class of examples is when the curvature of $L$ is fibrewise positive (or more generally of constant signature $n_-$) for some metric on $L$. In this case $E^k_q$'s   are concentrated in the single degree $q=n_-$ by the appropriate Kodaira/Le Poitier vanishing theorems.\
But are there others? Perhaps when the $E^k_q$'s concentrate in multiple degrees?