Holomorphic Approximation of Symplectic Diffeomorphisms for Calogero--Moser Spaces

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 ${\mathcal{C}}_n$.