A few comments on (hyper)kähler geometry

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 Kähler manifold to be hyperkähler is $h_{i\bar{k}}{h}_{j\bar{l}}\hat{I}{\text{c◯}}^{\bar{k}\bar{l}}=C\hat{I}{\text{c◯}}_{ij}$, where $h_{i\bar{k}}$ is a complex metric, $\hat{I}\text{c◯}$ is a symplectic matrix and $C$ is a positive constant. $\bullet$ The procedure of Kähler reduction includes two stages. On the first stage, a Kähler manifold of dimension $2n$ is reduced to a $(2n-1)$ - dimensional manifold, while on the second stage, one arrives at a Kähler 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 hyperkähler reduction of ${\mathbb{R}}^7\times S^1$ down to the Taub-NUT metric.

Minor corrections. A reference added