Fixed points of symplectic periodic flows

TL;DR

This paper proves that symplectic circle actions with certain conditions have at least 1 + dim(M)/2 fixed points, using equivariant cohomology, and shows no symplectic flows have only one or two fixed points in high dimensions.

Contribution

It extends classical fixed point results to broader symplectic actions using ABBV localization and provides new lower bounds on fixed points without assumptions.

Findings

At least 1 + dim(M)/2 fixed points under certain conditions.

No symplectic periodic flow with 1 or 2 fixed points in dimensions ≥8.

Uses equivariant cohomology to generalize classical results.

Abstract

The study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least 1 + dim(M)/2 fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah-Bott-Berline-Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective -- the Chern class map assigns to a fixed point the sum of the action weights at the point. We complement this result with less sharp lower bounds on the number of fixed points, under no assumptions; from a dynamical systems viewpoint, our results imply that there is no symplectic periodic flow with exactly one or two equilibrium points on a compact manifold of dimension at least eight.