Certain circle actions on Kaehler manifolds

Abstract

Let the circle act holomorphically on a compact K\"ahler manifold $M$ of complex dimension $n$ with moment map $\phi\colon M\to\R$. Assume the critical set of $\phi$ consists of 3 connected components, the extrema being isolated points. We show that $M$ is equivariantly biholomorphic to $\CP^n$, where $n\geq 2$, or to $\Tilde G_2(\R^{n+2})$, the Grassmannian of oriented 2-planes in $\R^{n+2}$, where $n\geq 3$, with a standard circle action; we also show that $M$ is equivariantly symplectomorphic to $\CP^n$, where $n\geq 2$, or to $\Tilde G_2(\R^{n+2})$, where $n\geq 3$, with a standard circle action.