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> **Question.** Is it known/easy to see that every smooth projective variety $X$ (over an algebraically closed field), except for the point and $\mathbb{P}^1$, has a vector bundle which is not a direct sum of line bundles? 
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I have a result which trivially shows the above fact in positive characteristic, but I don't know whether I should state it as a corollary as it might be well-known or obvious for reasons I don't see. 



Existence of non-split vector bundles on smooth projective varieties
Yes, it is true that (over an algebraically closed field) the only positive-dimensional smooth projective variety on which every algebraic vector bundle splits as a sum of line bundles is $\mathbb P^1$.  

More generally [Ballico has proved](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&ved=0CEAQFjAB&url=http%3A%2F%2Frendiconti.dmi.units.it%2Fvolumi%2F37%2F10.pdf&ei=rlVPUYiuLsWtO-iBgeAJ&usg=AFQjCNE5zdU1a7Sr8pyEJDq5yruOmgHcCQ&bvm=bv.44158598,d.ZWU) that if every algebraic vector bundle splits as a sum of line bundles on a reduced *but maybe reducible* positive-dimensional projective variety , then that variety is a chain of $\mathbb P^1$'s.  

For holomorphically inclined readers, [here](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CDcQFjAA&url=http%3A%2F%2Fwww.m-hikari.com%2Fimf-password%2F1-4-2006%2FballicoIMF1-4-2006-3.pdf&ei=q0VPUefAMMbOhAfO1oCYAQ&usg=AFQjCNFAOjN-49cdM77_EFDPc9rIwe2JPw&bvm=bv.44158598,d.ZG4) is a related result in complex geometry.


