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Direct Products for the Hamiltonian Density Property

Rafael B. Andrist, Gaofeng Huang

TL;DR

The paper proves that HDP is preserved under direct products of Stein (and affine) manifolds, linking HDP to SDP when $H^1_{dR}(X)=0$. It then establishes HDP (and SDP in favorable cases) for key spaces: $(\mathbb{C}^*)^{2n}$ and the traceless Calogero–Moser spaces $\mathcal{S}_n$, with a detailed Poisson-bracket analysis for $\mathcal{S}_n$ and a constructive, induction-based proof that the Hamiltonian generators recover the full invariant Poisson algebra. As applications, the results yield a Carleman-type approximation for Hamiltonian diffeomorphisms on a real form of $\mathcal{S}_n$ and extend the density of holomorphic symplectic automorphisms to Calogero–Moser–type spaces. Together, these findings illuminate the structure of holomorphic symplectic automorphism groups and provide tools for approximation in complex symplectic geometry.

Abstract

We show that the direct product of two Stein manifolds with the Hamiltonian density property enjoys the Hamiltonian density property as well. We investigate the relation between the Hamiltonian density property and the symplectic density property. We then establish the Hamiltonian and the symplectic density property for $(\mathbb{C}^\ast)^{2n}$ and for the so-called traceless Calogero--Moser spaces. As an application we obtain a Carleman-type approximation for Hamiltonian diffeomorphisms of a real form of the traceless Calogero--Moser space.

Direct Products for the Hamiltonian Density Property

TL;DR

The paper proves that HDP is preserved under direct products of Stein (and affine) manifolds, linking HDP to SDP when . It then establishes HDP (and SDP in favorable cases) for key spaces: and the traceless Calogero–Moser spaces , with a detailed Poisson-bracket analysis for and a constructive, induction-based proof that the Hamiltonian generators recover the full invariant Poisson algebra. As applications, the results yield a Carleman-type approximation for Hamiltonian diffeomorphisms on a real form of and extend the density of holomorphic symplectic automorphisms to Calogero–Moser–type spaces. Together, these findings illuminate the structure of holomorphic symplectic automorphism groups and provide tools for approximation in complex symplectic geometry.

Abstract

We show that the direct product of two Stein manifolds with the Hamiltonian density property enjoys the Hamiltonian density property as well. We investigate the relation between the Hamiltonian density property and the symplectic density property. We then establish the Hamiltonian and the symplectic density property for and for the so-called traceless Calogero--Moser spaces. As an application we obtain a Carleman-type approximation for Hamiltonian diffeomorphisms of a real form of the traceless Calogero--Moser space.

Paper Structure

This paper contains 9 sections, 26 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Direct Products
  3. Applications
  4. $\mathbb{C}^2$
  5. $\mathbb{C} \times \mathbb{C}^\ast$
  6. The torus $(\mathbb{C}^\ast)^{2n}$
  7. Traceless Calogero--Moser space
  8. Poisson brackets for $\mathcal{S}_n$
  9. Proof of Hamiltonian Density Property for $\mathcal{S}_n$

Key Result

Theorem 2.1

Let $(X, \omega_X)$ and $(Y, \omega_Y)$ be smooth affine varieties equipped with algebraic symplectic forms. If $(X, \omega_X)$ and $(Y, \omega_Y)$ have the algebraic Hamiltonian density property, then $(X \times Y, \omega_{X \times Y})$ has the algebraic Hamiltonian density property.

Theorems & Definitions (55)

  • Definition 1.1
  • Remark 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof : Proof of Theorem \ref{['thm:directprodHam']}
  • proof : Proof of Theorem \ref{['thm:directprodHamStein']}
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 45 more