On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data

TL;DR

This paper investigates whether the isomorphism type of semi-free Hamiltonian $S^1$-manifolds in six dimensions can be determined by fixed point data, refining previous assumptions and applying $J$-holomorphic methods.

Contribution

It introduces additional conditions on the reduced spaces and fixed surfaces, enabling the use of $J$-holomorphic techniques to classify these manifolds.

Findings

Counterexamples to Gonzales' assumptions

Additional assumptions suffice for classification

Application to positive monotone symplectic manifolds

Abstract

Following Gonzales, we answer the question of whether the isomorphism type of a semi-free Hamiltonian $S^1$-manifold of dimension six is determined by certain data on the critical levels. We first give counter examples showing that Gonzales' assumptions are not sufficient for a positive answer. Then we prove that it is enough to further assume that the reduced spaces of dimension four are symplectic rational surfaces and the interior fixed surfaces are restricted to at most one level. The additional assumptions allow us to use results proven by $J$-holomorphic methods. Gonzales' answer was applied by Cho in proving that if the underlying symplectic manifold is positive monotone then the space is isomorphic to a Fano manifold with a holomorphic $S^1$-action. We show that our variation is enough for Cho's application.