Irreducible holomorphic bundles on the twistor space of a simple hyperkähler manifold are generically fibrewise stable when their rank is 2 or 3, or when they are simple on some fibre.
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Families of stable bundles on the fibres of the hyperk\"ahler twistor projection
Irreducible holomorphic bundles on the twistor space of a simple hyperkähler manifold are generically fibrewise stable when their rank is 2 or 3, or when they are simple on some fibre.