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

Abstract

There is the close relation between smooth $SU(p,q)$-actions on ${\mathbf P}^{p+q-1}_{\mathbb C}$ and triples of smooth functions satisfying four conditions. To construct smooth $SU(p,q)$-actions on ${\mathbf P}^{p+q-1}_{\mathbb C}$, we construct triples of smooth functions satisfying four conditions. As a result, we can show that for given positive integer $m$ there exist uncountably infinite equivalence classes of smooth $SU(p,q)$-actions on ${\mathbf P}^{p+q-1}_{\mathbb C}$ with $m$ closed and $m+1$ open orbits, and furthermore we have new smooth $SU(p,q)$-actions on $S^{2p+2q-1}$ with $m$ closed and $m+1$ open orbits.