Certain circle actions on Kaehler manifolds

TL;DR

This paper classifies compact Kähler manifolds with a holomorphic circle action and exactly three critical components of the moment map, showing they are biholomorphic or symplectomorphic to standard models like complex projective space or Grassmannians.

Contribution

It provides a classification of such manifolds under the given symmetry and critical set conditions, identifying them with well-known geometric spaces.

Findings

Manifolds are biholomorphic to P^n or G_2(\u00bb^{n+2})

Manifolds are symplectomorphic to P^n or G_2(bb^{n+2})

Classification holds for na0bgeqa02 and na0bgeqa03 respectively

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.