Hamiltonian circle actions with fixed point set almost minimal

TL;DR

This paper classifies certain Hamiltonian circle actions on symplectic manifolds with fixed point sets almost minimal, showing they are uniquely determined and equivalent to standard Grassmannian examples under specific conditions.

Contribution

It characterizes Hamiltonian circle actions with almost minimal fixed point sets, establishing their uniqueness and equivalence to Grassmannian models under cohomology and Kähler conditions.

Findings

The fixed point set configuration determines the manifold's topology and Chern classes.

Such actions are uniquely modeled by the Grassmannian of oriented 2-planes.

Kähler and holomorphic conditions imply biholomorphic and symplectomorphic equivalence.

Abstract

Motivated by recent works on Hamiltonian circle actions satisfying certain minimal conditions, in this paper, we consider Hamiltonian circle actions satisfying an almost minimal condition. More precisely, we consider a compact symplectic manifold $(M, \omega)$ admitting a Hamiltonian circle action with fixed point set consisting of two connected components $X$ and $Y$ satisfying $\dim(X)+\dim(Y)=\dim(M)$. Under certain cohomology conditions, we determine the circle action, the integral cohomology rings of $M$, $X$ and $Y$, and the total Chern classes of $M$, $X$, $Y$, and of the normal bundles of $X$ and $Y$. The results show that these data are unique --- they are exactly the same as those in the standard example $\Gt_2(\R^{2n+2})$, the Grassmannian of oriented $2$-planes in $\R^{2n+2}$, which is of dimension $4n$ with (any) $n\in\N$, equipped with a standard circle action. Moreover, if $M$ is K\"ahler and the action is holomorphic, we can use a few different criteria to claim that $M$ is $S^1$-equivariantly biholomorphic and $S^1$-equivariantly symplectomorphic to $\Gt_2(\R^{2n+2})$.