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Periodic symplectic and Hamiltonian diffeomorphisms on irrational ruled surfaces

Nicholas Lindsay, Weiyi Zhang

TL;DR

This work addresses extending finite symplectic/Cyclic actions to Hamiltonian circle actions on irrational ruled $4$-manifolds. It develops a fibered framework using $J$-holomorphic sphere fibrations and invariant almost complex structures to analyze group actions, uncovering obstructions for involutions and positive extension results for higher-order cyclic actions. The authors construct exotic, non-extendable symplectic involutions (including Klein four-group examples) and prove that, for $k>2$, homologically trivial $oldsymbol{Z}_k$-actions extend to Hamiltonian $S^1$-actions in key settings (e.g., $M\cong\Sigma_g\times S^2$ and minimal ruled surfaces with $b_2=2$). They also classify finite symplectic group actions via fibrations, describe fixed-point sets, and provide algebraic exemplars informed by Maruyama’s theory of automorphism groups, linking symplectic topology with complex-analytic automorphisms.

Abstract

We study the extension of homologically trivial symplectic or Hamiltonian cyclic actions to Hamiltonian circle actions on irrational ruled symplectic $4$-manifolds. On one hand, we construct symplectic involutions on minimal irrational ruled $4$-manifolds that cannot extend to a symplectic circle action even with a possibly different symplectic form. Higher dimensional examples are also constructed. On the other hand, for homologically trivial symplectic cyclic actions of any other order, we show that such an extension always exists. We also classify finite groups of symplecticmorphisms that acts trivially on the first homology group, and prove the non-extendability of the Klein $4$-group action to the three dimensional rotation group action motivated by the classification of finite groups of symplectomorphisms.

Periodic symplectic and Hamiltonian diffeomorphisms on irrational ruled surfaces

TL;DR

This work addresses extending finite symplectic/Cyclic actions to Hamiltonian circle actions on irrational ruled -manifolds. It develops a fibered framework using -holomorphic sphere fibrations and invariant almost complex structures to analyze group actions, uncovering obstructions for involutions and positive extension results for higher-order cyclic actions. The authors construct exotic, non-extendable symplectic involutions (including Klein four-group examples) and prove that, for , homologically trivial -actions extend to Hamiltonian -actions in key settings (e.g., and minimal ruled surfaces with ). They also classify finite symplectic group actions via fibrations, describe fixed-point sets, and provide algebraic exemplars informed by Maruyama’s theory of automorphism groups, linking symplectic topology with complex-analytic automorphisms.

Abstract

We study the extension of homologically trivial symplectic or Hamiltonian cyclic actions to Hamiltonian circle actions on irrational ruled symplectic -manifolds. On one hand, we construct symplectic involutions on minimal irrational ruled -manifolds that cannot extend to a symplectic circle action even with a possibly different symplectic form. Higher dimensional examples are also constructed. On the other hand, for homologically trivial symplectic cyclic actions of any other order, we show that such an extension always exists. We also classify finite groups of symplecticmorphisms that acts trivially on the first homology group, and prove the non-extendability of the Klein -group action to the three dimensional rotation group action motivated by the classification of finite groups of symplectomorphisms.

Paper Structure

This paper contains 18 sections, 38 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homologically trivial cyclic group actions on closed orientable surfaces of positive genus
  4. Almost complex structures
  5. Weights of cyclic group actions
  6. Automorphism groups of irrational ruled surfaces
  7. Background on Question \ref{['kedra']} and related work
  8. Finite symplectic actions on irrational ruled $4$-manifolds
  9. Construction of the fibration
  10. Classification of finite groups acting on irrational ruled symplectic $4$-manifolds
  11. Fixed point sets of symplectic cyclic actions
  12. Exotic symplectic involutions on irrational ruled symplectic $4$-manifolds
  13. Exotic actions of the Klein 4-group on irrational ruled symplectic $4$-manifolds
  14. Extension of finite symplectic actions on irrational ruled $4$-manifolds
  15. Algebraic Examples
  16. ...and 3 more sections

Key Result

Theorem 1.2

There exist homologically trivial symplectic involutions on irrational ruled $4$-manifolds with base genus at least two and any $b_2\ge 2$, such that they cannot be extended to symplectic circle actions, even after modifying the symplectic form.

Figures (3)

  • Figure 1: The complete root system of Lagrangian spheres with the induced homological action
  • Figure 2: The involution on $\tilde{S}$ and the invariant circle $C$.
  • Figure 3: The geometric construction of the embeddings $i_{1},i_{2}$ and $i_{3}$.

Theorems & Definitions (71)

  • Theorem 1.2
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • proof
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 2.1
  • ...and 61 more