Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c\_1=1, c\_2=6, c\_3=0$.



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



Dual of stable vector bundle on a Fano threefold
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 $E^\* \cong E(-1)$.


