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