Symplectic reduction of Sasakian manifolds

TL;DR

This paper extends symplectic reduction techniques from Kähler to Sasakian manifolds, providing a new framework for understanding group actions and quotients in Sasakian geometry.

Contribution

It generalizes the symplectic reduction process to Sasakian manifolds under semisimple group actions, bridging a gap between Kähler and Sasakian geometries.

Findings

Established a reduction procedure for Sasakian manifolds analogous to Kähler reduction.

Identified the quotient of a Sasakian manifold by a semisimple group action with a specific geometric construction.

Extended the categorical quotient concept to the Sasakian setting.

Abstract

When a complex semisimple group $G$ acts holomorphically on a K\"ahler manifold $(X,\omega)$ such that a maximal compact subgroup $K\subset G$ preserves the symplectic form $\omega$, a basic result of symplectic geometry says that the corresponding categorical quotient $X/G$ can be identified with quotient of the zero-set of the moment map by the action of $K$. We extend this to the context of a semisimple group acting on a Sasakian manifold.