On toric hyperkähler manifolds with compact complex submanifolds

Abstract

A toric hyperkähler manifold is defined as a hyperkähler quotient of the flat quaternionic space H^N by a subtorus of the real torus T^N. The purposes of this paper are to construct compact complex submanifolds of toric hyperkähler manifolds, and to show that our hyperkähler manifold is a resolution of singularities of an affine algebro-geometric quotient. We also show that these submanifolds are biholomorphic to Delzant spaces, which are Kähler quotients of C^N by subtori of T^N. Finally, we apply these results to determining whether complex structures on our hyperkähler manifold are equivalent.