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 $q=0$ (or more generally $q=n_-$) by the appropriate Kodaira type vanishing theorems.

But are there other examples? Perhaps when the $E^k_q$'s concentrate in multiple degrees?