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

Abstract

Abstract 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.