Hyperholomorpic connections on coherent sheaves and stability

Abstract Let M be a hyperkähler manifold, and F a torsion-free and reflexive coherent sheaf on M . Assume that F (outside of singularities) admits a connection ∇ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [V2]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M . In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessary L-integrable. We show that such sheaves are polystable.