Hamiltonian circle actions with minimal isolated fixed points

TL;DR

This paper investigates Hamiltonian circle actions on compact symplectic manifolds with exactly n+1 isolated fixed points, revealing that such manifolds are characterized by specific weights and are isomorphic to known examples like complex projective space or certain Grassmannians.

Contribution

It establishes a complete characterization of manifolds with minimal isolated fixed points under Hamiltonian circle actions, linking weights to cohomology and Chern classes, and identifying them with classical examples.

Findings

Fixed points count is at least n+1 for Hamiltonian circle actions.

Manifolds with exactly n+1 fixed points are determined by a particular weight.

Such manifolds are isomorphic to P^n or t_2(\u00a2^{n+2}) with standard actions.

Abstract

Let the circle act in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$ of dimension $2n$. Then the $S^1$-action has at least $n+1$ fixed points. We study the case when the fixed point set consists of precisely $n+1$ isolated points. We first show certain equivalence on the first Chern class of $M$ and some particular weight of the $S^1$-action at some fixed point. Then we show that the particular weight can completely determine the integral cohomology ring of $M$, the total Chern class of $M$, and the sets of weights of the $S^1$-action at all the fixed points. We will see that all these data are isomorphic to those of known examples, $\CP^n$, or $\Gt_2(\R^{n+2})$ with $n\geq 3$ odd, equipped with standard circle actions.