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On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data

Liat Kessler, Nikolas Wardenski

TL;DR

The paper constructs explicit counterexamples showing that fixed point data alone do not determine the isomorphism type of six-dimensional semi-free Hamiltonian S^1-manifolds, challenging Go11. It then establishes a refined local-to-global framework under two key hypotheses: that all four-dimensional reduced spaces are symplectic rational surfaces and that interior fixed components occur at at most one level. With these assumptions, the authors develop the Morse-flow toolkit, define almost symplectic μ-S^1-diffeomorphisms, and prove extension results across critical levels, enabling a global isomorphism result in many cases and preserving Cho’s classification outcomes for positive monotone six-dimensional semi-free S^1-manifolds. The work blends Morse theory, J-holomorphic/symplectic techniques, and rigidity results to bridge local data at critical levels with global isomorphism types, while highlighting the limitations of prior rigidity assertions in Go11.

Abstract

Following Gonzales, we answer the question of whether the isomorphism type of a semi-free Hamiltonian $S^1$-manifold of dimension six is determined by certain data on the critical levels. We first give counter examples showing that Gonzales' assumptions are not sufficient for a positive answer. Then we prove that it is enough to further assume that the reduced spaces of dimension four are symplectic rational surfaces and the interior fixed surfaces are restricted to at most one level. The additional assumptions allow us to use results proven by $J$-holomorphic methods. Gonzales' answer was applied by Cho in proving that if the underlying symplectic manifold is positive monotone then the space is isomorphic to a Fano manifold with a holomorphic $S^1$-action. We show that our variation is enough for Cho's application.

On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data

TL;DR

The paper constructs explicit counterexamples showing that fixed point data alone do not determine the isomorphism type of six-dimensional semi-free Hamiltonian S^1-manifolds, challenging Go11. It then establishes a refined local-to-global framework under two key hypotheses: that all four-dimensional reduced spaces are symplectic rational surfaces and that interior fixed components occur at at most one level. With these assumptions, the authors develop the Morse-flow toolkit, define almost symplectic μ-S^1-diffeomorphisms, and prove extension results across critical levels, enabling a global isomorphism result in many cases and preserving Cho’s classification outcomes for positive monotone six-dimensional semi-free S^1-manifolds. The work blends Morse theory, J-holomorphic/symplectic techniques, and rigidity results to bridge local data at critical levels with global isomorphism types, while highlighting the limitations of prior rigidity assertions in Go11.

Abstract

Following Gonzales, we answer the question of whether the isomorphism type of a semi-free Hamiltonian -manifold of dimension six is determined by certain data on the critical levels. We first give counter examples showing that Gonzales' assumptions are not sufficient for a positive answer. Then we prove that it is enough to further assume that the reduced spaces of dimension four are symplectic rational surfaces and the interior fixed surfaces are restricted to at most one level. The additional assumptions allow us to use results proven by -holomorphic methods. Gonzales' answer was applied by Cho in proving that if the underlying symplectic manifold is positive monotone then the space is isomorphic to a Fano manifold with a holomorphic -action. We show that our variation is enough for Cho's application.

Paper Structure

This paper contains 9 sections, 39 theorems, 150 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Application to Cho's classification of positive monotone semi-free Hamiltonian $S^1$-manifolds
  3. Non-isomorphic manifolds with the same fixed point data
  4. Preliminaries: the Morse flow
  5. Almost symplectic $\mu-S^1$-diffeomorphisms of free Hamiltonian $S^1$-manifolds
  6. Isomorphisms of neighborhoods of critical levels
  7. Extending an isomorphism over a critical level
  8. Local Data
  9. Proofs of results in \ref{['sec:almostsymplectic']}

Key Result

Theorem 1.9

For $i=1,2$, let $M^i=(M^i,\omega^i,\mu_i)$ and $\lambda$ be as in set:intro. Assume that $M^1$ and $M^2$ have the same $*$-small fixed point data at the critical value $\lambda$, and that $\lambda$ is the only critical value of $\mu_i$ for $i=1,2$. Assume that Let $r>0$ be such that there is no critical value in $[\lambda-r,\lambda)$ with respect to both $\mu_1$ and $\mu_2$. Consider an isomorph

Figures (8)

  • Figure 1: On the left: the images of the fixed points of $N$ under $\mu_N$. On the right: the toric momentum image of $N$, where the red lines represent the level sets $\lambda$ and $\lambda'$ of $\mu_N$.
  • Figure 2: On the left, the momentum polytope of the con-compact symplectic toric manifold we started with, on the right the momentum polytope of the same manifold after twisting the action.
  • Figure 3:
  • Figure 4: The $T^2$-momentum images of the reduced spaces at $y=0$ for both open symplectic toric manifolds. The red lines correspond to the fixed spheres in the respective symplectic toric manifolds and are of the same length. The arrows indicate how the cross section changes as $y$ decreases.
  • Figure 5: The preimage of the red region under the momentum map from the toric symmetric $S^2\times S^2$ to $\mathbb R^2$ is the embedded closed ball $\mathcal{B}$.
  • ...and 3 more figures

Theorems & Definitions (96)

  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.9
  • Theorem 1.10
  • proof : Proof of \ref{['thm:mainresult']}, assuming \ref{['thm:extending-g']}
  • Conjecture 1.11
  • Lemma 1.12
  • Theorem 1.13
  • Theorem 1.14
  • ...and 86 more