Let $X$ and $Y$ be connected quasi-projective varieties over $\mathbf{C}$. Let $\mathcal{L}$ be an algebraic vector bundle over $X\times Y$. Let $p_2:X\times Y\rightarrow Y$ be the projection.

($\star$) Assume that for all $y_0\in Y$ one has that $\mathcal{L}|_{X\times \{y_0\}}$ is trivial.

In this case, the first (naive) intuition is that $\mathcal{L}\simeq p_2^*\mathcal{F}$ for some algebraic vector bundle $\mathcal{F}$ over $Y$.

**Q1**: Give an example where $(\star)$ is satified but where $\mathcal{L}$ is not the pullback
of any algebraic vector bundle $\mathcal{F}$ over $Y$.

**Q2**: Under what (interesting) additional assumptions on $X$, $Y$ and $\mathcal{L}$ is it possible to conclude that $\mathcal{L}$ is the pullback of a vector bundle over $Y$?

For example, if $X$ is complete then the square theorem says that $\mathcal{L}$ is the pullback of a vector bundle over $Y$.