The rigidity of Dolbeault-type operators and symplectic circle actions

TL;DR

This paper proves the rigidity of Dolbeault-type operators on compact almost-complex manifolds and uses this to derive criteria for Hamiltonian symplectic circle actions with isolated fixed points, simplifying and generalizing existing results.

Contribution

It establishes the rigidity of Dolbeault-type operators in this setting and applies it to characterize Hamiltonian circle actions, extending previous results.

Findings

Rigidity of Dolbeault-type operators on almost-complex manifolds.

Derived identities for weights at fixed points of circle actions.

Provided criteria to determine Hamiltonian circle actions.

Abstract

Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it gives a criterion to detemine whether or not a symplectic circle action with isolated fixed points is Hamiltonian. As applications, we simplify the proofs of some known results related to symplectic circle actions, due to Godinho, Tolman-Weitsman and Pelayo-Tolman, and generalize some of them to more general cases.