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On the Hofer-Zehnder conjecture on weighted projective spaces

Simon Allais

Abstract

We prove an extension of the homology version of the Hofer-Zehnder conjecture proved by Shelukhin to the weighted projective spaces which are symplectic orbifolds. In particular, we prove that if the number of fixed points counted with their isotropy order as multiplicity of a non-degenerate Hamiltonian diffeomorphism of such a space is larger than the minimum number possible, then there are infinitely many periodic points.

On the Hofer-Zehnder conjecture on weighted projective spaces

Abstract

We prove an extension of the homology version of the Hofer-Zehnder conjecture proved by Shelukhin to the weighted projective spaces which are symplectic orbifolds. In particular, we prove that if the number of fixed points counted with their isotropy order as multiplicity of a non-degenerate Hamiltonian diffeomorphism of such a space is larger than the minimum number possible, then there are infinitely many periodic points.
Paper Structure (15 sections, 23 theorems, 104 equations)

This paper contains 15 sections, 23 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The category of weighted projective spaces
  4. Projective join
  5. Generating functions of ${\mathbb{R}^*_+\times S^1}$-equivariant Hamiltonian diffeomorphisms
  6. Generating functions homology
  7. Action and generating functions
  8. Homology of sublevel sets and local homology of a fixed point
  9. Composition morphisms and the direct system of $G_*^{(a,b)}({\boldsymbol{\sigma}})$
  10. Properties of the generating functions homology
  11. Smith inequality
  12. $\mathbb{Z}/p\mathbb{Z}$-action of a $p$-iterated generating function
  13. Application of Smith inequality and computation of $\beta_{\mathrm{tot}}$
  14. Proof of the main theorem
  15. Homology properties of orbibundles

Key Result

Theorem 1.1

Every Hamiltonian diffeomorphism $\varphi$ of $\mathbb{C}\mathrm{P}({\mathbf{q}})$ satisfies

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2: Bor60
  • Lemma 2.3
  • Proposition 3.1
  • Corollary 3.2
  • Proposition 3.3
  • ...and 20 more