Suppose I have a vector bundle $V$ over a smooth projective variety $X$ with the following exotic property: any global section of $V$ does not vanish anywhere.
The general question what can we say about this vector bundle? For example, it looks like it is not necessary generated by global sections: it could be not enough global sections. But can we claim that it has a free direct summand of rank $h^0(X, V)$?
But mostly I'm interested in tangent bundle $V=T_X$. Could someone give me an example of $X$ s.t. $T_X$ has this property, but is not trivial as a vector bundle? Is this possible?