Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions

TL;DR

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

Contribution

It establishes a classification of these symplectic manifolds as Fano manifolds with specific holomorphic actions, including a comprehensive list and description of all semifree * actions.

Findings

Identifies all six-dimensional monotone symplectic manifolds with semifree circle actions as Fano manifolds.

Provides a complete list of such Fano manifolds and their semifree * actions.

Shows these manifolds are -equivariantly symplectomorphic to specific Fano manifolds.

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 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.