Holomorphic Approximation of Symplectic Diffeomorphisms for Calogero--Moser Spaces

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 ${\mathcal{C}}_n$.

arXiv commentAdjusted title and introduction