An equivariant description of certain holomorphic symplectic varieties

TL;DR

This paper demonstrates that under certain conditions, holomorphic symplectic varieties with Hamiltonian G-actions are isomorphic to product varieties of a complex semisimple group and a regular Slodowy slice, revealing a canonical structure in integrable systems.

Contribution

The paper proves that such varieties are G-equivariantly isomorphic to G times a regular Slodowy slice, providing a classification result for these integrable systems.

Findings

Holomorphic symplectic varieties with Hamiltonian G-actions are isomorphic to G×S_reg under certain conditions.

Establishes a canonical form for abstract integrable systems arising from Hamiltonian G-actions.

Connects geometric structures with integrable systems in hyperkähler and physical contexts.

Abstract

This short note considers varieties of the form $G\times S_{\text{reg}}$, where $G$ is a complex semisimple group and $S_{\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$. Such varieties arise naturally in hyperk\"ahler geometry, theoretical physics, and in the theory of abstract integrable systems developed by Fernandes, Laurent-Gengoux, and Vanhaecke. In particular, previous work of the author and Rayan uses a Hamiltonian $G$-action to endow $G\times S_{\text{reg}}$ with a canonical abstract integrable system. One might therefore wish to understand, in some sense, all examples of abstract integrable systems arising from Hamiltonian $G$-actions. Accordingly, we consider a holomorphic symplectic variety $X$ carrying an abstract integrable system induced by a Hamiltonian $G$-action. Under certain hypotheses, we show that there must exist a $G$-equivariant variety isomorphism $X\cong G\times S_{\text{reg}}$.