Generators of cohomology groups of higher push-forward sheaves

Question

Let $\phi:S\rightarrow \mathbb{P}^1$ be an elliptic fibration of a compact complex surface. Assume that there is a multiple section $s$ of $\phi$. Is it true that $H^0(\mathbb{P}^1,R^2f_*\mathbb{R})$ is generated by the class of section $s$ and $H^2(\mathbb{P}^1,f_*\mathbb{R})$ is generated by the fiber class of $\phi$? I would appreciate it if someone could introduce a good reference to me.

Answer 1

The first claim can be wrong if the elliptic surface has singular fibers with multiple components. This increases $R^2$ at those points, giving it a punctual subsheaf, giving it more global sections. We can associate these with the classes of fibral divisors that have zero intersection with the class of a fiber, say.

Your second claim follows from the general fact that, as long as the fibers are geometrically connected, the lowest filtered piece of the Leray spectral sequence comes from functoriality of cohomology, so the classes in it are all pullbacks.

I don't know a reference.