Mathematical Research Letters

Volume 10 (2003)

Number 4

Hyperkähler manifolds with torsion obtained from hyperholomorphic bundles

Pages: 501 – 513



Misha Verbitsky (Glasgow University)


We construct examples of compact hyperkähler manifolds with torsion (HKT manifolds) which are not homogeneous and not locally conformal hyperkähler. Consider a total space $T$ of a tangent bundle over a hyperkähler manifold $M$. The manifold $T$ is hypercomplex, but it is never hyperkähler, unless $M$ is flat. We show that $T$ admits an HKT-structure. We also prove that a quotient of $T$ by a $\Bbb Z$-action $v \arrow q^n v$ is HKT, for any real number $q\in \Bbb R$, $q >1$. 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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