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Non-extendablity of Shelukhin's quasimorphism and non-triviality of Reznikov's class

Morimichi Kawasaki, Mitsuaki Kimura, Shuhei Maruyama, Takahiro Matsushita, Masato Mimura

TL;DR

This work investigates the extendability of Shelukhin's quasimorphism $\mathfrak{S}_M$ on $\widetilde{\mathrm{Ham}^c}(M,\omega)$ to $\widetilde{\mathrm{Symp}_0^c}(M,\omega)$ for selected high-dimensional symplectic manifolds, proving non-extendability for products $M=(S,\omega_S)\times(N,\omega_N)$ with genus$(S)\ge2$ and for blow-ups of tori. The authors relate this to Reznikov's characteristic class $R$, proving its nontriviality on these manifolds, and they connect these phenomena to relative bounded cohomology and flux considerations, highlighting differences between symplectic and volume-preserving groups. A key technical ingredient is the computation of the average Hermitian scalar curvature after blow-ups, $A(\hat{M}_r,\omega_\rho)\neq0$, which feeds into obstruction arguments via Shelukhin's construction. The results illuminate intrinsic obstructions to extending invariant quasimorphisms and establish new instances of Reznikov class nontriviality, with implications for foliated bundle theory and the geometry of diffeomorphism groups.

Abstract

Shelukhin constructed a quasimorphism on the universal covering of the group of Hamiltonian diffeomorphisms for a general closed symplectic manifold. In the present paper, we prove the non-extendability of that quasimorphism for certain symplectic manifolds, such as a blow-up of torus and the product of a surface of genus at least two and a closed symplectic manifold. As its application, we prove the non-vanishing of Reznikov's characteristic class for the above symplectic manifolds.

Non-extendablity of Shelukhin's quasimorphism and non-triviality of Reznikov's class

TL;DR

This work investigates the extendability of Shelukhin's quasimorphism on to for selected high-dimensional symplectic manifolds, proving non-extendability for products with genus and for blow-ups of tori. The authors relate this to Reznikov's characteristic class , proving its nontriviality on these manifolds, and they connect these phenomena to relative bounded cohomology and flux considerations, highlighting differences between symplectic and volume-preserving groups. A key technical ingredient is the computation of the average Hermitian scalar curvature after blow-ups, , which feeds into obstruction arguments via Shelukhin's construction. The results illuminate intrinsic obstructions to extending invariant quasimorphisms and establish new instances of Reznikov class nontriviality, with implications for foliated bundle theory and the geometry of diffeomorphism groups.

Abstract

Shelukhin constructed a quasimorphism on the universal covering of the group of Hamiltonian diffeomorphisms for a general closed symplectic manifold. In the present paper, we prove the non-extendability of that quasimorphism for certain symplectic manifolds, such as a blow-up of torus and the product of a surface of genus at least two and a closed symplectic manifold. As its application, we prove the non-vanishing of Reznikov's characteristic class for the above symplectic manifolds.

Paper Structure

This paper contains 20 sections, 34 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Backgrounds of our main results
  3. Main results
  4. Notation and conventions
  5. Organization of the present paper
  6. Preliminaries
  7. Symplectic geometry
  8. Quasimorphisms
  9. Group cohomology and bounded cohomology
  10. Shelukhin's quasimorphism
  11. Reznikov class
  12. Symplectic blow-up
  13. Average Hermitian scalar curvature of the blow-up of a torus
  14. Proof of Theorem \ref{['main_non_ext']}
  15. Recapitulation of our constructions of symplectomorphisms
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $(S,\omega)$ be a closed connected orientable surface whose genus is at least two with a symplectic form $\omega$. Then, Py's Calabi quasimorphism $\mu_P\colon \mathrm{Ham}^c(S,\omega) \to \mathbb{R}$ is not extendable to $\mathrm{Symp}^c_0(S,\omega)$.

Figures (2)

  • Figure 1: The vector fields $X_a$ and $Y_b$ defining $\sigma_{\mathscr{A}}$ and $\tau_{\mathscr{A}}$ when $a>0$ and $b>0$.
  • Figure 2: Idea of the proof of Proposition \ref{['prop:flux']} when $M$ is a closed surface of genus at least two and $N$ is the one-point set

Theorems & Definitions (61)

  • Definition 1.1
  • Theorem 1.2: KK
  • Theorem 1.3: KKMM
  • Theorem A: non-extendaility of Shelukhin's quasimorphism
  • Theorem B: non-triviality of the Reznikov class
  • Theorem C: comparison between symplectic preserving and volume-preserving
  • Corollary 1.4
  • Proposition 2.1: Ban, see also Ban97 and MS
  • Lemma 2.2
  • Lemma 2.3: homogenization of quasimorphisms
  • ...and 51 more