\(Sp(1)^n\)-invariant quaternionic Kähler metric (Q2715747)

The paper under review is a note about quaternionic Kähler manifolds of dimension \(4n\) with \(Sp(1)^n\)-symmetry. Hitchin reduced the construction of \(Sp(1)\)-invariant hyper-Kähler metrics on \(\mathbb{H}-\{ 0\}\) to a non-linear system of ordinary differential equations of first order for three functions. The first observation is that a generalization of the Hitchin ansatz to higher dimensions gives only the product examples. Then the moduli space \(M_k\) of self-dual connections on a rank 2 vector bundle over \(\mathbb{H}\) with second Chern class \(k\) is considered. It is a hyper-Kähler manifold of dimension \(8k\), the points of which can be represented by certain quaternion-valued matrices using the ADHM construction.NEWLINENEWLINENEWLINEImposing a trace zero condition in this matrix picture, defines a hyper-Kähler submanifold \(M_k^0 \subset M_k\) of dimension \(8k-4\). Using a subgroup of the conformal group of \(\mathbb{H}\) and of the gauge group at infinity, one can define an action of the group \(Sp(1)\times R^+\) on \(M_k^0\), which preserves the underlying quaternionic structure (but not the hypercomplex structure). The quaternionic Kähler reduction of \(M_k^0\) by that action is a quaternionic Kähler space \(P(M_k^0)\) with singularities, which has been considered by Boyer an Mann. In the case \(k=2\) the residual symmetry is still big enough such that \(P(M_2^0)\) is a quaternionic Kähler space of dimension 8 with \(Sp(1)^2\)-symmetry (whereas \(M_2^0\) is an example of a hyper-Kähler manifold of dimension 12 with \(Sp(1)^3\)-symmetry).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].