Hamiltonian circle actions with minimal fixed sets

TL;DR

This paper classifies compact symplectic manifolds with minimal fixed sets under Hamiltonian circle actions, showing they are topologically and cohomologically similar to complex projective spaces or certain Grassmannians.

Contribution

It proves that such manifolds are isomorphic in cohomology and Chern classes to either projective space or Grassmannian, and characterizes their fixed set data precisely.

Findings

Manifolds are cohomologically equivalent to ^n or Grassmannians.

Fixed set data matches standard circle actions on these manifolds.

No points with stabilizer k for k > 2.

Abstract

Consider an effective Hamiltonian circle action on a compact symplectic $2n$-dimensional manifold $(M, \omega)$. Assume that the fixed set $M^{S^1}$ is {\em minimal}, in two senses: it has exactly two components, $X$ and $Y$, and $\dim(X) + \dim(Y) = \dim(M) - 2$. We prove that the integral cohomology ring and Chern classes of $M$ are isomorphic to either those of $\CP^n$ or (if $n \neq 1$ is odd) to those of $\Gt_2(\R^{n+2})$, the Grassmannian of oriented two-planes in $\R^{n+2}$. In particular, $H^i(M;\Z) = H^i(\CP^n;\Z)$ for all $i$, and the Chern classes of $M$ are determined by the integral cohomology {\em ring}. We also prove that the fixed set data of $M$ agrees exactly with the fixed set data for one of the standard circle actions on one of these two manifolds. In particular, we show that there are no points with stabilizer $\Z_k$ for any $k > 2$. The same conclusions hold when $M^{S^1}$ has exactly two components and the even Betti numbers of $M$ are minimal, that is, $b_{2i}(M) = 1$ for all $i \in \{0,...,1/2\dim(M)\}$. This provides additional evidence that very few symplectic manifolds with minimal even Betti numbers admit Hamiltonian actions.