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A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite...
In this paper, we consider the problem of determining which topological complex rank-2
vector bundles on non-Kähler elliptic surfaces admit holomorphic structures; in
particular, we give necessary and sufficient conditions for the existence of holomorphic
rank-2 vector bundles on non-{Kä}hler elliptic surfaces.
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