Relative Kodaira Vanishing?

Question

Let $ X , S $ be proper varieties over $ \mathbb{C} $ and $ \pi : X \rightarrow S $ be a smooth, proper morphism with relative canonical line bundle $ K_{X/S} $. If $ L $ is a $ \pi $-relatively ample line bundle on $ X $, is there a class of examples where it is true that $ H^q(X, K_{X/S} \otimes L) = 0 $ for all $ q \ge 1 $?

Answer 1

The natural analogue of Kodaira vanishing is $R^q\pi_*(K_{X/S} \otimes \mathscr L) = 0$ for $q > 0$, which follows from Kodaira vanishing plus 'cohomology and base change' [Hartshorne, Thm. III.12.11(a)].

Thus the Leray spectral sequence for $\pi$ collapses on the $E_2$ page, giving $$H^q(X,K_{X/S} \otimes \mathscr L) \cong H^q\big(S,\pi_*(K_{X/S} \otimes \mathscr L)\big).$$ Then the question becomes a positivity question for $\pi_*(K_{X/S}\otimes \mathscr L)$. The situation is underdefined to answer this: the $\pi$-ampleness hypothesis on $\mathscr L$ is preserved by tensoring with $\pi^* \mathscr M$ for any line bundle $\mathscr M$ on $S$, and the projection formula gives $$\pi_*(K_{X/S}\otimes \mathscr L \otimes \pi^*\mathscr M) \cong \pi_*(K_{X/S} \otimes \mathscr L) \otimes \mathscr M.$$ In particular, taking $\mathscr M = \mathcal O_S(d)$ for $d \gg 0$ gives the required vanishing by Serre vanishing.

Note that $\mathscr L\otimes \pi^* \mathcal O_S(d)$ is ample for $d \gg 0$, but this doesn't seem to help. At any rate, the result cannot possibly be true in general without some sort of 'absolute' restriction on $\mathscr L$.