Serre-Swan's theorem (see the [MO discussion](https://mathoverflow.net/questions/36286/holomorphic-vector-bundles-and-swans-theorem)) says that any locally free sheaf over an affine variety is a direct summand of a free sheaf. However, this is not true on projective varieties. It is not hard to check that a non-trivial line bundle with non-zero global sections can not be a direct summand of a free sheaf. The reason is that, being a direct summand of a free sheaf implies that the dual line bundle also has non-zero global sections. But that implies the line bundle is trivial. 


I am wondering if the same holds for vector bundles, i.e. if a vector bundle and its dual over a projective variety both have non-zero global sections, then the vector bundle is trivial. 


Another I think related question is the following:


Is a direct summand of a direct sum of line bundles on a projective variety also a direct sum of line bundles?



Direct summands of direct sum of line bundles on projective varieties
1) The vector bundle $E=\mathcal O(1) \oplus \mathcal O(-1)$ over $\mathbb P^1$ has non trivial sections and so has its dual (which happens to be the same bundle $E$ ).  

 However $E$ is non trivial because each of its global sections has a zero. 


2) Yes, a direct summand of a direct sum of line bundles *on a complete variety* is also a direct sum of line bundles.  

This follows from [Atiyah's general version of the Krull-Schmidt theorem](http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1956__84_/BSMF_1956__84__307_0/BSMF_1956__84__307_0.pdf) . 


**Edit**  

Since Fei asks, let me remark that the trick in 1) works also for bundles which are not direct sums of line bundles:  

If $T$ is the tangent bundle to $\mathbb P^2$, then $E=T \oplus T^\* $ has non-trivial global sections , and so has $E^\*=E$  

However, since $T$ is indecomposable $E$ is not a sum of line bundles by Atiyah's result and is thus *a fortiori* not trivial.


