Slices for reductive group actions in algebraic and holomorphic symplectic geometry

TL;DR

This paper extends symplectic slice theorems to complex reductive algebraic groups and Hamiltonian symplectic groupoid actions, broadening the understanding of local structures in algebraic and holomorphic symplectic geometry.

Contribution

It introduces new definitions of Poisson and symplectic slices and proves their analogues for complex reductive algebraic groups and symplectic groupoid actions.

Findings

Analogues of classical symplectic slice theorems for complex reductive algebraic groups.

Use of Slodowy slices and decomposition classes in complex Lie algebras.

Generalization of symplectic slice theorems to Hamiltonian symplectic groupoid actions.

Abstract

Symplectic slice theorems elucidate the local structure of symplectic manifolds carrying Hamiltonian actions of compact Lie groups. We generalize these theorems in two natural settings. The first is based on the idea that complex reductive algebraic groups are the natural complex-geometric counterparts of compact Lie groups. Using new definitions of Poisson and symplectic slices, we prove analogues of the classical symplectic slice theorems for Hamiltonian actions of complex reductive algebraic groups. These analogues have versions in the complex-algebraic and holomorphic categories, and make extensive use of Slodowy slices and decomposition classes in complex reductive Lie algebras. The starting point for our second setting is the fact that Hamiltonian Lie group actions are special cases of Hamiltonian symplectic groupoid actions. We generalize the classical symplectic slice theorems to the latter case.