Constructions of smooth \(SU(p, q)\)-actions on the projective space \(\mathbf P^{p + q - 1}_{\mathbb C}\) with \(m\) closed and \(m + 1\) open orbits
The authors consider actions of generalized special unitary groups on complex projective spaces. Assume \(p,q\geq 3\) and \(m>0\). The main result says that there exist uncountably many equivalence classes of smooth \(SU(p,q)\)-actions on \({\mathbf{P}}^{p+q-1}_{\mathbb{C}}\) with \(m\) closed and \(m+1\) open orbits. It then follows that there also exist uncountably many equivalence classes of smooth \(SU(p,q)\)-actions on the sphere \({\mathbb{S}}^{2p+2q-1}\) with \(m\) closed and \(m+1\) open orbits. The proof is based on constructing triples of real valued smooth functions on a circle satisfying specific conditions.
scientific article; zbMATH DE number 7227183