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The symplectic holomorphic density property for Calogero-Moser spaces

Rafael B. Andrist, Gaofeng Huang

TL;DR

The paper defines and develops the symplectic holomorphic density property and the Hamiltonian holomorphic density property for Stein manifolds, and proves that Calogero–Moser spaces $\mathcal{C}_n$ possess both properties. By combining invariant theory of matrix tuples, rank-one reductions, and detailed bracket calculations, it is shown that a finite set of Hamiltonians ($\operatorname{tr} Y$, $\operatorname{tr} Y^2$, $\operatorname{tr} X^3$, $(\operatorname{tr} X)^2$) generates all holomorphic Hamiltonians on $\mathcal{C}_n$, enabling an Andersén–Lempert-type approximation of any symplectic holomorphic automorphism by complete flows. For $n=2$ and general $n$, the authors construct explicit generators and brackets, culminating in a full description of the identity component of the symplectic holomorphic automorphism group as the closure of flows from these Hamiltonians. The results connect Calogero–Moser spaces to Hilbert schemes and extend known density results from $T^*\mathbb{C}^n$ to a rich, higher-dimensional setting with strong transitivity properties.

Abstract

We introduce the symplectic holomorphic density property and the Hamiltonian holomorphic density property together with the corresponding version of Andersén-Lempert theory. We establish these properties for the Calogero-Moser space $\mathcal{C}_n$ of $n$ particles and describe its group of holomorphic symplectic automorphisms.

The symplectic holomorphic density property for Calogero-Moser spaces

TL;DR

The paper defines and develops the symplectic holomorphic density property and the Hamiltonian holomorphic density property for Stein manifolds, and proves that Calogero–Moser spaces possess both properties. By combining invariant theory of matrix tuples, rank-one reductions, and detailed bracket calculations, it is shown that a finite set of Hamiltonians (, , , ) generates all holomorphic Hamiltonians on , enabling an Andersén–Lempert-type approximation of any symplectic holomorphic automorphism by complete flows. For and general , the authors construct explicit generators and brackets, culminating in a full description of the identity component of the symplectic holomorphic automorphism group as the closure of flows from these Hamiltonians. The results connect Calogero–Moser spaces to Hilbert schemes and extend known density results from to a rich, higher-dimensional setting with strong transitivity properties.

Abstract

We introduce the symplectic holomorphic density property and the Hamiltonian holomorphic density property together with the corresponding version of Andersén-Lempert theory. We establish these properties for the Calogero-Moser space of particles and describe its group of holomorphic symplectic automorphisms.
Paper Structure (11 sections, 39 theorems, 107 equations)

This paper contains 11 sections, 39 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Hamiltonian and Symplectic Holomorphic Density Property
  3. Warm-Up: $T^\ast \mathbb{C}^n$
  4. The Calogero--Moser space
  5. The ring of invariant functions
  6. Proof of the Hamiltonian Density Property for $n=2$
  7. Finding $\mathop{\mathrm{tr}}\nolimits(X^k) \cdot \mathop{\mathrm{tr}}\nolimits(Y^l)$
  8. Finding $\mathop{\mathrm{tr}}\nolimits(X^i Y^j) \cdot \mathop{\mathrm{tr}}\nolimits(X^k Y^l)$
  9. Finding a smaller set of generators
  10. Brackets between generating invariant functions
  11. Proof of the Hamiltonian Density Property for $n > 2$

Key Result

Theorem 3

The Calogero--Moser space $\mathcal{C}_n$, $n \in \mathbb{N}$, has the Hamiltonian holomorphic density property.

Theorems & Definitions (84)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Corollary 4
  • proof
  • Theorem 5
  • Corollary 6
  • proof
  • Definition 7
  • Remark 8
  • ...and 74 more