Cotangent bundles of toric varieties and coverings of toric hyperk\"ahler manifolds

Toric hyperk{\"a}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{\"a}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{\"a}hler manifold are sufficient to glue the desired toric hyperk{\"a}hler manifold.