Hyperholomorphic sheaves and new examples of hyperkaehler manifolds

Abstract

Given a compact hyperkaehler manifold $M$ and a holomorphic bundle B over $M$, we consider a Hermitian connection $\nabla$ on B which is compatible with all complex structures on $M$ induced by the hyperkaehler structure. Such a connection is unique, because it is Yang-Mills. We call the bundles admitting such connections hyperholomorphic bundles. A stable bundle is hyperholomorphic if and only if its Chern classes $c_1$, $c_2$ are SU(2)-invariant, with respect to the natural SU(2)-action on the cohomology. For several years, it was known that the moduli space of stable hyperholomorphic bundles is singular hyperkaehler. More recently, it was proven that singular hyperkaehler varieties admit a canonical hyperkaehler desingularization. In the present paper, we show that a moduli space of stable hyperholomorphic bundles is compact, given some assumptions on Chern classes of B and hyperkaehler geometry of $M$ (we also require $dim_C M>2$). Conjecturally, this leads to new examples of hyperkaehler manifolds. We develop the theory of hyperholomorphic sheaves, which are (intuitively speaking) coherent sheaves compatible with hyperkaehler structure. We show that hyperholomorphic sheaves with isolated singularities can be canonically desingularized by a blow-up. This theory is used to study degenerations of hyperholomorphic bundles.