Feix-Kaledin metric on the total spaces of cotangent bundles to K\"ahler quotients

TL;DR

This paper explores the geometry of cotangent bundle spaces over K"ahler quotients, constructing hyperk"ahler metrics via reduction techniques and analyzing their completeness and algebraic structures.

Contribution

It constructs a hyperk"ahler metric on cotangent bundles of K"ahler quotients and analyzes the metric's completeness and algebraic properties of the space.

Findings

The hyperk"ahler metric coincides with the Feix-Kaledin metric.

The metric completion forms a stratified hyperk"ahler space.

A necessary condition for metric completeness is provided.

Abstract

In this paper we study the geometry of the total space $Y$ of a cotangent bundle to a K\"ahler manifold $N$ where $N$ is obtained as a K\"ahler reduction from $\mathbb C^n$. Using the hyperk\"ahler reduction we construct a hyperk\"ahler 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 hyperk\"ahler 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 $Y = T^*N$ 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.