Singular Reduction of Generalized Complex Manifolds

TL;DR

This paper extends the theory of symplectic reduction to generalized complex and Kähler manifolds, showing that group actions produce well-structured quotients with generalized complex geometry.

Contribution

It establishes an analogue of singular reduction for Hamiltonian generalized complex and Kähler manifolds, generalizing classical symplectic reduction results.

Findings

Partition of the quotient by orbit types yields generalized complex manifolds.

Results apply to Hamiltonian actions of compact Lie groups.

Extension to generalized Kähler manifolds demonstrated.

Abstract

In this paper, we develop results in the direction of an analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a generalized complex manifold in a Hamiltonian fashion, then the partition of the global quotient by orbit types induces a partition of the Lin-Tolman quotient into generalized complex manifolds. This result holds also for reduction of Hamiltonian generalized K\"ahler manifolds.