*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?

allbundles on $\mathbf{CP}^n$ are linearly trivial, so then you are saying all bundles on $\mathbf{CP}^n$ split? Or did I misunderstand? $\endgroup$ – Mahdi Majidi-Zolbanin Jan 31 '13 at 21:52