A few comments on (hyper)k\"ahler geometry

TL;DR

This paper provides explicit conditions for a Kähler manifold to be hyperkähler and discusses the process of Kähler and hyperkähler reduction with illustrative examples.

Contribution

It offers a simple explicit proof of the necessary and sufficient condition for hyperkählerity and clarifies the two-stage Kähler reduction process with examples.

Findings

Derived a simple explicit condition for hyperkähler manifolds.

Clarified the two-stage Kähler reduction process.

Illustrated reduction procedures with models like $ ext{R}^3 imes S^1$ and Taub-NUT.

Abstract

In this note, we make two methodical observations. $\bullet$ We prove in a simple explicit way that a necessary and sufficient condition for a K\"ahler manifold to be hyperk\"ahler is $h_{i\bar k} h_{j\bar l } \Omega^{\bar k \bar l} \ =\ C \Omega_{ij}$, where $h_{i\bar k}$ is a complex metric, $\Omega$ is a symplectic matrix and $C$ is a positive constant. $\bullet$ The procedure of K\"ahler reduction includes two stages. On the first stage, a K\"ahler manifold of dimension $2n$ is reduced to a $(2n-1)$ - dimensional manifold, while on the second stage, one arrives at a K\"ahler manifold of dimension $2(n-1)$. We note that this second stage has the meaning of Hamiltonian reduction. We illustrate the procedure by discussing a simple toy model when $\mathbb{R}^3 \times S^1$ is reduced down to $S^2$. We elucidate also hyperk\"ahler reduction of $\mathbb{R}^7 \times S^1$ down to the Taub-NUT metric.