Unimodality of the Betti numbers for Hamiltonian circle actions with index-increasing moment maps

TL;DR

This paper proves the unimodality of Betti numbers for Hamiltonian circle actions on compact symplectic manifolds of any dimension, under the condition that the moment map is index-increasing, extending previous results in specific dimensions.

Contribution

It generalizes the proof of the unimodality conjecture to all dimensions for Hamiltonian circle actions with index-increasing moment maps.

Findings

Unimodality of Betti numbers holds in arbitrary dimensions under the index-increasing assumption.

The approach extends equivariant cohomology techniques to higher dimensions.

The conjecture is confirmed for a broader class of symplectic manifolds.

Abstract

The unimodality conjecture posed by Tolman in the conference `Moment maps in Various Geometry" in 2005 states that if (M,w) is a 2n-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers is unimodal. Recently, the author and M. Kim proved that the unimodality holds in eight-dimensional cases by using equivariant cohomology theory. In this paper, we generalize the idea in \cite{CK} to an arbitrary dimensional case. Also, we prove the conjecture in arbitrary dimension with an assumption that a moment map "index-increasing."