Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions

Let $(M,ω_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that $(M,ω_M)$ is $S^1$-equivariant symplectomorphic to some Kähler Fano manifold $(X,ω_X, J)$ with a certain holomorphic $\mathbb{C}^*$-action. We also give a complete list of all such Fano manifolds and describe all semifree $\mathbb{C}^*$-actions on them specifically.

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