Hamiltonian circle actions on eight dimensional manifolds with minimal fixed sets

TL;DR

This paper proves that for 8-dimensional symplectic manifolds with a Hamiltonian circle action and minimal fixed points, a key positivity condition always holds, confirming a conjecture about their topological and symplectic structure.

Contribution

It establishes that the positivity condition always holds in this setting, completing the proof of the symplectic Petrie conjecture for these manifolds.

Findings

Positivity condition always holds for these manifolds.

Cohomology rings and Chern classes match those of linear actions on .

Confirms the symplectic Petrie conjecture in this case.

Abstract

Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic manifold $M$ with exactly five fixed points, which is the smallest possible fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$ satisfies an extra "positivity condition" then the isotropy weights at the fixed points of $M$ agree with those of some linear action on $\mathbb{CP}^4$. Therefore, the (equivariant) cohomology rings and the (equivariant) Chern classes of $M$ and $\mathbb{CP}^4$ agree; in particular, $H^*(M;\mathbb{Z}) \simeq \mathbb{Z}[y]/y^5$ and $c(TM) = (1+y)^5$. In this paper, we prove that this positivity condition always holds for these manifolds. This completes the proof of the "symplectic Petrie conjecture" for Hamiltonian circle actions on on 8-dimensional closed symplectic manifolds with minimal fixed sets.