Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions

Abstract

Let $(M,{\omega }_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that $(M,{\omega }_M)$ 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.

Complete version (with 110 pages and many figures). We have splitted this manuscript into three parts arXiv:1812.09892, arXiv:1904.10962, and arXiv:1905.07292