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Let $C$ be a smooth complex projective curve and let $E$ be a rank two vector bundle over $C$. If $E$ is decomposable, ie $E=L\oplus M$ for some line bundles $L$ and $M$ we have that $$ E^\vee \otimes \det(E) = (L^\vee\oplus M^\vee)\otimes L\otimes M = M\oplus L \simeq E. $$ My question is: Under which conditions does it happen to an indecomposable $E$?

More generally, If $E$ is an indecomposable vector budle of arbitrary rank over a compact complex manifold $X$. Under which conditions $E\simeq E^\vee \otimes L$ for some line bundle $L$?

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For any rank 2 vector bundle there is a pairing $$ E \otimes E \to \det(E). $$ Being non-degenerate it induces an isomorphism $E \cong E^\vee \otimes \det(E)$, which thus holds for any rank 2 vector bundle.

For the more general question --- such an isomorphism exists if and only if there is a non-degenerate pairing $E \otimes E \to L$.

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