Calculation with Leray spectral sequence The Leray spectral sequence is a cohomological spectral sequence of the form $$H^p(Y;R^q f_*(F)) \Longrightarrow H^{p+q}(X;F)$$
for abelian sheaves $F$ on a site $X$ and morphisms of sites $f : X \to Y$. Is there an example of a concrete calculation with the Leray spectral sequence for sheaf cohomology?  So far I have "only" seen abstract and general arguments which use the Leray spectral sequence; my question is not about these general usages. Often the spectral sequence degenerates directly (at least, in the examples I am aware of), which is not very interesting and doesn't show the real power of spectral sequences. Actually I guess that these cases of the Leray spectral sequence may be replaced by more "direct" arguments.
The cohomological Serre spectral sequence associated to a Serre fibration follows from the Lerre spectral sequence and in algebraic topology there are lots of calculations with the Serre spectral sequence. So I am actually asking for calculations with the Lerre spectral sequence which rather belong to sheaf theory and are not instances of the Serre spectral sequence.
 A: As per Lee's suggestion, take a smooth projective morphism $f:X \to Y$ with the base a genus 2 curve and fibers elliptic curves. From a theorem of Blanchard-Deligne, the $E_2$-page of the Leray-Serre spectral sequence degenerates, hence giving the isomorphism
$$
H^k(X;\underline{\mathbb{Q}}_X) \cong \bigoplus_{k = p + q} H^p(Y;\mathbf{R}^qf_*(\underline{\mathbb{Q}}_X))
$$
And if you'd like, the $E_2$ page of with no monodromy in the cohomology looks like
\begin{align*}
\begin{matrix}
H^0(Y; \underline{\mathbb{Q}}_Y) & H^1(Y; \underline{\mathbb{Q}}_Y) & H^2(Y; \underline{\mathbb{Q}}_Y)\\
H^0(Y; \underline{\mathbb{Q}}_Y\oplus \underline{\mathbb{Q}}_Y) & H^1(Y; \underline{\mathbb{Q}}_Y\oplus \underline{\mathbb{Q}}_Y) & H^2(Y;\underline{\mathbb{Q}}_Y\oplus \underline{\mathbb{Q}}_Y)\\
H^0(Y; \underline{\mathbb{Q}}_Y) & H^1(Y; \underline{\mathbb{Q}}_Y) & H^2(Y; \underline{\mathbb{Q}}_Y)
\end{matrix} & = 
\begin{matrix}
\mathbb{Q} & \mathbb{Q}^{\oplus 4} & \mathbb{Q}\\
 \mathbb{Q}\oplus \mathbb{Q}& \mathbb{Q}^{\oplus 4} \oplus \mathbb{Q}^{\oplus 4} & \mathbb{Q}\oplus \mathbb{Q}\\
\mathbb{Q} & \mathbb{Q}^{\oplus 4} & \mathbb{Q}
\end{matrix}
\end{align*}
