AN EQUIVARIANT DESCRIPTION OF CERTAIN HOLOMORPHIC SYMPLECTIC VARIETIES

摘要

Varieties of the form $Gimes S_{!ext{reg}}$ , where $G$ is a complex semisimple group and $S_{!ext{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$ , arise naturally in hyperk?hler geometry, theoretical physics and the theory of abstract integrable systems. Crooks and Rayan [‘Abstract integrable systems on hyperk?hler manifolds arising from Slodowy slices’, Math. Res. Let., to appear] use a Hamiltonian $G$ -action to endow $Gimes S_{!ext{reg}}$ with a canonical abstract integrable system. To understand examples of abstract integrable systems arising from Hamiltonian $G$ -actions, 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 $Xcong Gimes S_{!ext{reg}}$ .