Let $\phi:S\rightarrow \mathbb{CP}^1$ be an elliptic fibration of a K3 surface. When is the Leray spectral sequence associated to the fibration $E_2$degenerate? Are there any good criteria for the $E_2$degeneration?
It's always true with $\mathbb{Q}$ coefficients. It follows from a general result of Zucker, who proved Leray degenerates whenever you have a projective map to curve. But in this case, it's simpler to check it by hand. There is only 

