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Centralizers of Hamiltonian circle actions on rational ruled surfaces

Pranav Chakravarthy, Martin Pinsonnault

TL;DR

The paper classifies the homotopy type of centralizers of Hamiltonian S^1 actions on rational ruled surfaces, showing a dichotomy between a torus-based centralizer and a pushout of two tori along S^1, depending on toric extensions of the action. It blends Delzant and Karshon classifications with J-holomorphic curve techniques to analyze the action on the space of invariant almost complex structures and proves that invariant strata correspond to toric extensions, yielding a concrete pushout description and a product with ΩS^3 in the nontrivial two-orbits case. The results rely on stratifications of the invariant J-space, equivariant Gromov-type theorems, and detailed isotropy representations for even and odd Hirzebruch surfaces, with explicit homological and cohomological computations (including Pontryagin algebras) to identify the homotopy type of centralizers. Overall, the work provides a complete picture of how Hamiltonian circle actions sit inside the symplectomorphism groups of these four-manifolds and establishes tools applicable to finite abelian actions as well.

Abstract

In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$ under the presence of Hamiltonian group actions of the circle $S^1$. We prove that the group of equivariant symplectomorphisms are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. This follows from the analysis of the action of equivariant symplectomorphisms on the space of compatible and invariant almost complex structures $\mathcal{J}^{S^1}_ω$. In particular, we show that this action preserves a decomposition of $\mathcal{J}^{S^1}_ω$ into strata which are in bijection with toric extensions of the circle action. Our results rely on $J$-holomorphic techniques, on Delzant's classification of toric actions and on Karshon's classification of Hamiltonian circle actions on $4$-manifolds.

Centralizers of Hamiltonian circle actions on rational ruled surfaces

TL;DR

The paper classifies the homotopy type of centralizers of Hamiltonian S^1 actions on rational ruled surfaces, showing a dichotomy between a torus-based centralizer and a pushout of two tori along S^1, depending on toric extensions of the action. It blends Delzant and Karshon classifications with J-holomorphic curve techniques to analyze the action on the space of invariant almost complex structures and proves that invariant strata correspond to toric extensions, yielding a concrete pushout description and a product with ΩS^3 in the nontrivial two-orbits case. The results rely on stratifications of the invariant J-space, equivariant Gromov-type theorems, and detailed isotropy representations for even and odd Hirzebruch surfaces, with explicit homological and cohomological computations (including Pontryagin algebras) to identify the homotopy type of centralizers. Overall, the work provides a complete picture of how Hamiltonian circle actions sit inside the symplectomorphism groups of these four-manifolds and establishes tools applicable to finite abelian actions as well.

Abstract

In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of and under the presence of Hamiltonian group actions of the circle . We prove that the group of equivariant symplectomorphisms are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. This follows from the analysis of the action of equivariant symplectomorphisms on the space of compatible and invariant almost complex structures . In particular, we show that this action preserves a decomposition of into strata which are in bijection with toric extensions of the circle action. Our results rely on -holomorphic techniques, on Delzant's classification of toric actions and on Karshon's classification of Hamiltonian circle actions on -manifolds.
Paper Structure (45 sections, 115 theorems, 218 equations, 10 figures, 6 tables)

This paper contains 45 sections, 115 theorems, 218 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Hamiltonian torus actions on ruled 4-manifolds
  3. Preliminaries on Hamiltonian actions
  4. Torus actions on S2xS2 and CP2#CP2
  5. Symplectic forms on S2xS2 and CP2#CP2
  6. Toric actions on S2xS2 and CP2#CP2
  7. Hamiltonian circle actions on S2xS2 and CP2#CP2
  8. Equivalent circle actions
  9. Equivalent circle actions in Tm
  10. Toric extensions of circle actions
  11. Action of Symp(S2xS2) on JS1
  12. J-Holomorphic Preliminaries
  13. J-holomorphic spheres in S2xS2 and CP2#CP2
  14. Intersection of JS1 with the strata
  15. Symplectic actions of compact abelian groups on R4
  16. ...and 30 more sections

Key Result

Lemma 2.4

Let $(M, \omega)$ be a compact symplectic manifold equipped with a Hamiltonian torus action $\rho:T\times M\to M$ with moment map $\mu$. Then $\phi \in C(T)$ if, and only if, $\mu \circ \phi = \mu$ and $\phi \in \mathop{\mathrm{Symp}}\nolimits(M,\omega)$.

Figures (10)

  • Figure 1: Even Hirzebruch polygon
  • Figure 2: Odd Hirzebruch polygon
  • Figure 3: Graphs of circle actions with non-isolated fixed points
  • Figure 4: Graphs of circle actions whose fixed points are isolated
  • Figure 5: Graphs of $S^{1}(1,b;m)$ with possible locations of $C_{n}$ and $C_{-n}$
  • ...and 5 more figures

Theorems & Definitions (200)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • Remark 2.8
  • ...and 190 more