On isomorphisms of semi-free Hamiltonian S¹-manifolds and fixed point data

2025/05/20 by Liat Kessler, Kessler, Liat, Nikolas Wardenski +1 · 1 citation
Mathematics · #Geometry and complex manifolds #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2505.14000

Abstract

Following Gonzales, we answer the question of whether the isomorphism type of a semi-free Hamiltonian S¹-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 S¹-action. We show that our variation is enough for Cho's application.

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