Work over $\mathbb{C}$. Let $\Phi : X \to S$ be an elliptic fiber space, where $X$ is a smooth projective threefold with $H^1(\mathcal{O}_X)=H^2(\mathcal{O}_X)=0$, and $S$ is a smooth projective surface.
I want to show that $H^1(\mathcal{O}_S)=H^2(\mathcal{O}_S)=0$.
The vanishing of $H^1(\mathcal{O}_S)=0$ from $H^1(\mathcal{O}_X)=0$ can be achieved by a Leray spectral sequence argument, but a more direct argument would be more desirable.
