Feix-Kaledin metric on the total spaces of cotangent bundles to Kähler quotients

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 $\tilde{Y}$ of the space $Y$ is equipped with a structure of a stratified hyperkähler space. 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 $\tilde{Y}$ induced from quaternions. Suppose that $J\ne \pm I$ where $I$ is the complex structure whose restriction to $Y=T^{\ast }N$ is induced by the complex structure on $N$. We prove that the space ${\tilde{Y}}_J$ admits an algebraic structure and is an affine variety.

34 pages; fixed two mistakes (see Prop. 2.16 and Subs. 3.1) insignificant to the proof of the main results, improved some statements, added examples