Holomorphic Approximation of Symplectic Diffeomorphisms for Calogero--Moser Spaces
Gaofeng Huang
TL;DR
This paper proves that every symplectic diffeomorphism of the real Calogero–Moser space $\mathcal{C}_n^{\mathbb{R}}$, isotopic to the identity, can be approximated in the fine Whitney topology by holomorphic symplectic automorphisms of the complex Calogero–Moser space $\mathcal{C}_n$ that preserve $\mathcal{C}_n^{\mathbb{R}}$. The key strategy combines a refined $\tau$-symmetric density property for $\mathcal{C}_n$ with Carleman-type approximation techniques and an Andersén–Lempert–style local-to-global push-out, enabling local approximation on Saturn-like subsets and global assembly. Central to the approach is the identification of $\mathcal{C}_n^{\mathbb{R}}$ as the fixed-point set of the antiholomorphic involution $\tau$ and the demonstration that holomorphic vector fields generated by a small finite set of invariants suffice to approximate broader real Hamiltonian dynamics. The results extend the scope of Carleman and density-property methods from affine or coadjoint-orbit settings to the more intricate Calogero–Moser geometry, with potential implications for complex symplectic geometry and real-analytic approximations in nontrivial complexifications.
Abstract
The real Calogero--Moser space $\mathcal{C}_n^\mathbb{R}$ is a noncompact, totally real submanifold of the complex Calogero--Moser space $\mathcal{C}_n$. We prove that every symplectic diffeomorphism of $\mathcal{C}_n^\mathbb{R}$ smoothly isotopic to the identity can be approximated in the fine Whitney topology -- the strongest in this context -- by holomorphic symplectic automorphisms of $\mathcal{C}_n$ that preserve $\mathcal{C}_n^\mathbb{R}$. A key ingredient in our proof is a refined version of the symplectic density property of $\mathcal{C}_n$.