Cohomological localization for Hamiltonian $S^1$-actions and symmetries of complete intersections

TL;DR

This paper advances cohomological localization techniques for Hamiltonian circle actions, generalizing previous results and removing fixed point set assumptions, with applications to Betti number unimodality and symplectic rationality.

Contribution

It improves existing localization results by removing fixed point set assumptions and extends the analysis to higher-dimensional complete intersections.

Findings

Generalized cohomological localization results for Hamiltonian $S^1$-actions.

Removed fixed point set assumptions in dimension 8.

Connected localization results to Betti number properties and symplectic rationality.

Abstract

To begin the paper we revisit a cohomological localization result of Jones-Rawnsley which was subsequently improved by Farber, further generalizing the result. We then proceed to improve a previous result of the author on complete intersections of dimension $8k$ with a Hamiltonian $S^1$-action in two directions. Firstly, in dimension $8$ we remove the assumption on the fixed point set. Secondly, in any dimension we prove the result under an analogous assumption on the fixed point set. We also give some applications towards the unimodality of Betti numbers of symplectic manifolds having a Hamiltonian $S^1$-action, and discuss the relation to symplectic rationality problems.