Hyperholomorpic connections on coherent sheaves and stability

Details Hyperholomorpic connections on coherent sheaves and stability Hyperholomorpic connections on coherent sheaves and stability

2001-07-24

arxiv.org/abs/math/0107182v11

Cent. Eur. J. Math., 2011, 9(3), 535-557

Mathematics

Algebraic Geometry

37 pages, version 11, reference updated, corrected many minor errors and typos found by the referee

Scientific paper

10.2478/s11533-011-0016-0

Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singularities are hyperkaehler subvarieties in $M$. Such sheaves (called hyperholomorphic sheaves) are well understood. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessarily square-integrable. This situation arises often, for instance, when one deals with higher direct images of holomorphic bundles. We show that such sheaves are stable.

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