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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?

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If the fibers are compact Kähler manifolds, then the Hodge numbers of the fibers are constant and hence by Grauert's theorem, the direct images of the relative Hodge bundles are locally free (i.e., vector bundles). In other words, in this case $R^q\pi_*\Omega_W^p$ is a vector bundle for every $p,q\in \mathbb N$.

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