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Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c_1=1, c_2=6, c_3=0$.

Question. Is it true that $E(-1)=E^*$?

What I am able to show is that there is equality at the level of Chern characters, namely $\operatorname{ch}(E(-1))= \operatorname{ch}(E^*)$.

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For any vector bundle $E$ of rank 2 there is an isomorphism $$ E^* \cong E \otimes \det(E^*). $$ If $\det(E) = \mathcal{O}(1)$, this boils down to $E^* \cong E(-1)$.

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