Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions I

Abstract

Let $(M,{\omega }_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then $(M,{\omega }_M)$ is $S^1$-equivariant symplectomorphic to some Kähler Fano manifold $(X,{\omega }_X,J)$ with a certain holomorphic ${\mathbb{C}}^{\ast }$-action. We also give a complete list of all such Fano manifolds and describe all semifree ${\mathbb{C}}^{\ast }$-actions on them specifically.

arXiv comment52 pages, 22 figures