Calculation with Leray spectral sequence

Solution 1:

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*}