Holomorphic, torsion-parallel vector fields on KT manifolds induce Killing and holomorphic vector fields on Hermitian-Einstein moduli spaces, which become toric or QKT fibrations under extra closure conditions.
Hyperkaehler manifolds with torsion obtained from hyperholomorphic bundles
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We construct examples of compact hyperkaehler manifolds with torsion (HKT manifolds) which are not homogeneous and not locally conformal hyperkaehler. Consider a total space T of a tangent bundle over a hyperkaehler manifold M. The manifold T is hypercomplex, but it is never hyperkaehler, unless M is flat. We show that T admits an HKT-structure. We also prove that a quotient of T by a $\Z$-action $v \arrow q^n v$ is HKT, for any real number $q\in \R$, $q>1$. This quotient is compact, if M is compact. A more general version of this construction holds for all hyperholomorphic bundles with holonomy in Sp(n).
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Geometry and symmetries of Hermitian-Einstein and instanton connection moduli spaces
Holomorphic, torsion-parallel vector fields on KT manifolds induce Killing and holomorphic vector fields on Hermitian-Einstein moduli spaces, which become toric or QKT fibrations under extra closure conditions.