Symplectic capacities from Hamiltonian circle actions

TL;DR

This paper calculates symplectic capacities, specifically the Gromov width and Hofer-Zehnder capacity, for certain Fano symplectic manifolds with Hamiltonian circle actions using moment maps.

Contribution

It provides explicit computations of symplectic capacities for Fano manifolds with semifree Hamiltonian circle actions, a novel application of moment map techniques.

Findings

Gromov width of the manifold is explicitly computed.

Hofer-Zehnder capacity is determined for the class of manifolds.

Results connect symplectic capacities with Hamiltonian circle actions.

Abstract

Let $M$ be a closed Fano symplectic manifold with a semifree Hamiltonian circle action with isolated maximum. We compute the Gromov width and the Hofer-Zehnder capacity of $M$ using a moment map.