Let $X$ be a smooth projective variety and $E$ be a line bundle on $X$. Let $F$ be a line bundle on $X\times \mathbb{P}^1$. It is known that, if $p\_1^\*E|\_{X\_t}\cong F|\_{X\_t}$ for every $t\in \mathbb{P}^1$, then we have $F\cong E\boxtimes \mathcal{O}\_{\mathbb{P}^1}(k)$ for some $k$.



> 
> **Question.** Can this result be generalized? For example, does it hold when $E$ and $F$ are torsion-free sheaves, and $F$ is flat
> over $\mathbb{P}^1$?
> 
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Coherent sheaves that are isomorphic on every fibre
It seems to me that the statement is false already for rank $2$ vector bundles.


In fact, take $X=\mathbb{P}^1$. Then, it is known **[1]** that there exist indecomposable, uniform rank $2$ vector bundles $F$ on $\mathbb{P}^1 \times \mathbb{P}^1$, namely, indecomposable bundles such that $$F|\_{X\_t} = \mathcal{O}(a) \oplus \mathcal{O}(b)$$ for every $t \in \mathbb{P}^1$. So, setting $E=\mathcal{O}(a) \oplus \mathcal{O}(b)$, we have $$\pi\_1^\*E|\_{X\_t} = F|\_{X\_t}= \mathcal{O}(a) \oplus \mathcal{O}(b)$$ for every $t$. However, the two vector bundles $F$ and $E\boxtimes \mathcal{O}\_{\mathbb{P}^1}(k)$ can never be isomorphic, since the former is indecomposable whereas the latter is decomposable.


**References.**


**[1]** *Ballico, Edoardo*, Uniform vector bundles on quadrics, Ann. Univ. Ferrara, N. Ser., Sez. VII 27, 135-146 (1981). [ZBL0495.14008](https://zbmath.org/?q=an:0495.14008).


