The syzygy bundle S(E) of an initialized Ulrich bundle E on a smooth K3 surface or Fano variety X with index i_X > n-3 is semistable.
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Stability of syzygy bundles of Ulrich bundles
The syzygy bundle S(E) of an initialized Ulrich bundle E on a smooth K3 surface or Fano variety X with index i_X > n-3 is semistable.