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

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up vote 6 down vote accepted

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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Are you sure it is semi-stable? –  ginevra86 Feb 15 '11 at 14:28
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Every line bundle is semistable, and a direct sum of semistable sheaves with the same slope is semistable with that slope. –  Mattia Talpo Feb 15 '11 at 17:13

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