# Homologically trivial symplectic cyclic actions need not extend to Hamiltonian circle actions

## TL;DR

This paper presents examples of symplectic cyclic actions on four-manifolds that do not extend to Hamiltonian circle actions, challenging assumptions about the relationship between homologically trivial actions and Hamiltonian extensions.

## Contribution

It introduces holomorphic methods to analyze cyclic symplectic actions and demonstrates that trivial homology actions may not extend to Hamiltonian circle actions.

## Key findings

- Examples of non-extendable cyclic actions on four-manifolds.
- Application of holomorphic techniques to symplectic group actions.
- Extension of combinatorial tools to study cyclic symmetries.

## Abstract

We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to study cyclic actions.