Singular Reduction of Generalized Complex Manifolds

Abstract

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 Kähler manifolds.

LA-Courant algebroids and their applications

“…Remark 1.0.2. There are many interesting reduction procedures for both Poisson structures (see [25,26,40,70,77,83,89,124], and the references therein) and Courant algebroids (see [12,13,21,41,51,52,56,58,84,88,114,117,130], and the references therein). Regrettably, we will not have space to discuss all these treatments here.…”

“…Meanwhile, the LA-Dirac structure T g × d integrates to the multiplicative Dirac structure (g ⊕ g) × D ⊆ (q ⊕ q) × D. In this section, we apply the theory of LA-Dirac structures and VB-Dirac structures to the reduction of Courant algebroids. Reduction of Courant algebroids was first studied in [13,52,68,84,114,117,130], and further studied in [9,21,41,57,58,118].…”