Let $E$ be a locally free sheaf on a smooth projective variety $X$ over $\mathbb C$. Suppose that $\det E$ is an ample line bundle on $X$.
Is $$H^0(X,E^\vee) =0?$$
In fact, if $E$ of rank $1$, it is easy to see that $h^0(X,E^\vee) =h^0(X,\omega_X\otimes E)=0$ by Serre duality and Kodaira's vanishing theorem. (I think there should be an easier argument, though.)
If $E$ is of rank at least two, I thought about either
1) an induction argument, or
2) a strengthening of Kodaira's vanishing theorem for vector bundles (is there such a theorem?), or
3) an easy elementary argument that I'm simply not seeing...
Finally, if it helps, please do assume the canonical sheaf $\omega_X$ is ample.