*Recall a definition*. Let $V\subset \mathbb CP^n$ be a projective variety
and $E$ be a holomorphic vector bundle on it. We call $E$ *linearly trivial* if the restriction of $E$ to any projective line in $V$ is trivial.

It is well known that any linearly trivial bundle on $\mathbb CP^n$ itself is trivial (see Okonek, Schneider, Spindler). 
 
**Question 1.** I think that I have an idea of a generalization of this statement and would like
to ask you if this generalization is known?


**Generalized statement**. For any integer $n>0$ 
any linearly trivial bundle on any smooth degree $n$ hypersuface $V_n\subset \mathbb CP^{4n}$ is trivial. 

*Idea of the proof*. One can easily see that on $V_n$ any two points can be joined by a chain of two projective lines. Moreover for two points $x,y$ the set of such two-lines paths from $x$ to $y$ is a connected projective variety.
So let us trivialize the bundle at one point $x\in V_n$. Then extend this trivialization along each connected chain of $2$ lines on $V_n$ starting at $x$.
I think that the extension will be independent of the choice of a chain since 
the space of all chains from $x$ to $y$ is a connected projective variety, 
while all trivialization of $E$ over $y$ is an affine variety. 

**Question 2.** Does this reasoning sound plausible?