On toric hyperkähler manifolds with compact complex submanifolds

Abstract

A toric hyperkahler manifold is defined as a hyperkahler quotient of the flat quaternionic space HN by a subtorus of the real torus TN. The purposes of this paper are to construct compact complex submanifolds of toric hyperkahler manifolds, and to show that our hyperkahler 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 Kahler quotients of CN by subtori of TN. Finally, we apply these results to determining whether complex structures on our hyperkahler manifold are equivalent.