Real moment map and hyperkähler geometry techniques: the cotangent bundle to the 2-sphere and \(SL(2, \mathbb C)\)-adjoint orbits (Q2883144)

Following Kirwan and others, the author uses moment map techniques to study holomorphic actions of a complex reductive group \(G\) on a Kähler manifold \(M\). The approach uses a maximal compact subgroup \(G_u\) of \(G\), whose action on \(M\) is assumed to be Hamiltonian. Critical points of the moment maps are used to stratify \(M\) and to produce distinguished \(G_u\)-orbits inside \(G\)-orbits. In particular, following \textit{P. Heinzner, G. W. Schwarz} and \textit{H. Stötzel} [Compos. Math. 144, No. 1, 163--185 (2008; Zbl 1133.32009)], he relates the action of \(G\) and \(G_u\) with the action on \(M\) of a real form \(G_0\) of \(G\) and of \(K\), the complexification of \(K_0=G_0\cap G_u\). Heinzner, Schwarz and Stötzel, prove that, in the case of a flag manifold \(G/Q\), the orbits of the \(G_0\)- and \(K\)-actions satisfy Matsuki duality.NEWLINENEWLINEIn this paper, the author considers the case \(G=\mathrm{SL}(2,\mathbb{C})\), \(G_u=\mathrm{SU}(2)\), \(G_0=\mathrm{SU}(1,1)\) and \(K=\mathbb{C}^*\); he extends the above results to the case of the cotangent bundle to the sphere \(S^2\), which is described in three equivalent ways. Since the latter is a low-dimensional manifold, the author is able to provide an explicit, geometric description of the orbits, and to give some extensions of Matsuki duality to the case of interest.NEWLINENEWLINE The cotangent bundle of the sphere (and, more generally, of every compact Hermitian symmetric space) has a natural, \(G_u\)-invariant hyper-kähler structure. When a complex structure is fixed, the \(G_u\)-action extends to a holomorphic \(G\)-action, and the author describes how this holomorphic action changes as the complex structure (which is parametrized by a sphere \(S^2\)) varies.