A few comments on (hyper)kähler geometry
A. V. Smilga
TL;DR
The note characterizes when a Kähler manifold is hyperkähler by a multidimensional analogue of the heavenly equation, showing $h_{i\bar{k}} h_{j\bar{l}} \Omega^{\bar{k}\bar{l}} = C \Omega_{ij}$ (with $C>0$) is equivalent to hyperkähler structure, and provides an explicit construction of the corresponding complex structures with holonomy in $Sp(n)$. It then demonstrates Kähler reduction via a simple two-stage procedure, using a moment-map constraint to obtain a lower-dimensional Kähler metric and illustrating with a concrete toy model that leads to a sphere and preserves covariantly constant complex structure. The final part extends to hyperkähler reduction, applying the reduction to $\mathbb{R}^7 \times S^1$ to derive the Taub–NUT metric, with explicit moment maps $\mu_I,\mu_J,\mu_K$ and the resulting hyperkähler forms $\omega_I^{TN},\omega_J^{TN},\omega_K^{TN}$. Overall, the paper provides explicit, elementary methods to generate hyperkähler geometries from symmetry reductions and clarifies the role of moment maps in these constructions.
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 $h_{i\bar k} h_{j\bar l } Ω^{\bar k \bar l} \ =\ C Ω_{ij}$, where $h_{i\bar k}$ is a complex metric, $Ω$ 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.