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

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