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Does anybody know any example of a semistable and nonsimple sheaf on a K3 surface $S$?

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The trival sheaf $\mathcal O_S\oplus\mathcal O_S$ is one: it has non-trivial global endomorphism algebra $gl_2(k)$ and is semi-stable.

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  • $\begingroup$ Are you sure it is semi-stable? $\endgroup$
    – ginevra86
    Feb 15, 2011 at 14:28
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    $\begingroup$ Every line bundle is semistable, and a direct sum of semistable sheaves with the same slope is semistable with that slope. $\endgroup$
    – Mattia Talpo
    Feb 15, 2011 at 17:13

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