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In this paper, we study the geometry of the total space   $Y$   of a cotangent bundle to a Kähler manifold   $N$   where   $N$   is obtained as a Kähler reduction from   $\mathbb{C}^n$  . Using the hyperkähler reduction, we construct a hyperkähler metric on   $Y$   and prove that it coincides with the canonical Feix–Kaledin metric. This metric is in general non-complete. We show that the metric completion   $\widetilde Y$   of the space   $Y$   is equipped with a structure of a stratified hyperkähler space (in the sense of [  31 ]). We give a necessary condition for the Feix–Kaledin metric to be complete using an observation of R. Bielawski. Pick a complex structure   $J$   on   $\widetilde Y$   induced from the quaternions. Suppose that   $J\ne \pm I$   where   $I$   is the complex structure whose restriction to   $Y = T^{*}N$   is induced by the complex structure on   $N$  . We prove that the space   $\widetilde{Y}_J$   admits an algebraic structure and is an affine variety.

Feix–Kaledin Metric on the Total Spaces of Cotangent Bundles to Kähler Quotients