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

## TL;DR

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

## Contribution

It provides a classification of specific symplectic manifolds with semifree circle actions, linking them to Fano manifolds and explicitly describing their actions.

## Key findings

- Manifolds are symplectomorphic to Fano manifolds with holomorphic circle actions.
- Complete list of such Fano manifolds is provided.
- Explicit descriptions of the circle actions are given.

## 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 maximal and the minimal fixed component are both two dimensional, then $(M,\omega_M)$ is $S^1$-equivariantly symplectomorphic to some K\"{a}hler Fano manifold $(X, \omega_X, J)$ equipped with a certain holomorphic Hamiltonian $S^1$-action. We also give a complete list of all such Fano manifolds together with an explicit description of the corresponding $S^1$-actions.