For simplicity, assume everythings occur on a smooth projective variaty $X$.


Dual bundle of the given line bundle $\mathcal L$ is determined by $\mathcal L$ and $c\_1(\mathcal L)$.


$\mathcal L^\*= \mathcal L (-2c\_1(\mathcal L)) $


My question is, is their any similar relation between a vector bundle of rank >1 and its dual? Can dual vector bundle be described as a combination of original bundle and its data(e.g chern classes)? Even for a rank 2 case, I have no idea about this problem.


If you have one, please give me some short proof or sketch. Good reference is also very preferable. I appreciate any help.



dual of locally free sheaf
If $E$ is locally free of rank 2 then $E^\* \cong E(-c\_1(E))$. The isomorphism is induced by the nondegenerate pairing $E\otimes E \to \Lambda^2E \cong O(c\_1(E))$. For higher rank one can check that in general $E^\*$ is not a twist of $E$.


