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

## TL;DR

This paper classifies six-dimensional monotone symplectic manifolds with semifree circle actions, showing they are symplectomorphic to certain Fano manifolds with specific holomorphic actions, and provides a complete list of such manifolds.

## Contribution

It establishes a classification of these symplectic manifolds under given conditions and explicitly lists all corresponding Fano manifolds with semifree actions.

## Key findings

- Manifolds are symplectomorphic to Fano manifolds with holomorphic ${C}^*$-actions.
- Complete classification of such Fano manifolds is provided.
- All semifree ${C}^*$-actions on these manifolds are described explicitly.

## 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 K\"{a}hler Fano manifold $(X,\omega_X, J)$ 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.