The geometry and topology of toric hyperkahler manifolds

We study hyperkahler manifolds that can be obtained as hyperkahler quotients of flat quaternionic space by tori, and in particular, their relation to toric varieties and Delzant polytopes. When smooth, these hyperkahler quotients are complete. We also show that for smooth projective toric varieties X the cotangent bundle of X carries a hyperkahler metric, which is complete only if X is a product of projective spaces. Our hyperkahler manifolds have the homotopy type of a union of compact toric varieties intersecting along toric subvarieties. We give explicit formulas for the hyperkahler metric and its Kahler potential.