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

TL;DR

This paper explores the relationship between cotangent bundles of toric varieties and toric hyperkähler manifolds, establishing criteria for semi-stability and demonstrating how cotangent bundles can be used to construct hyperkähler manifolds.

Contribution

It provides a geometric framework for constructing toric hyperkähler manifolds from cotangent bundles of toric varieties using GIT quotients and semi-stability criteria.

Findings

Established geometric criteria for semi-stable points in GIT quotients.

Showed cotangent bundles of compact toric varieties suffice to glue toric hyperkähler manifolds.

Connected cotangent bundles with hyperkähler geometry through a gluing construction.

Abstract

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.