Reduction of Generalized Complex Structures

Abstract

We study reduction of generalized complex structures. More precisely, we investigate the following question. Let $J$ be a generalized complex structure on a manifold $M$, which admits an action of a Lie group $G$ preserving $J$. Assume that $M_0$ is a $G$-invariant smooth submanifold and the $G$-action on $M_0$ is proper and free so that $M_G:=M_0/G$ is a smooth manifold. Under what condition does $J$ descend to a generalized complex structure on $M_G$? We describe a sufficient condition for the reduction to hold, which includes the Marsden-Weinstein reduction of symplectic manifolds and the reduction of the complex structures in Kähler manifolds as special cases. As an application, we study reduction of generalized Kähler manifolds.

21 pages, definitive version

References in corpus (5)

• Reduction of Courant algebroids and generalized complex structures
• Symmetries in generalized Kähler geometry
• Hamiltonian symmetries and reduction in generalized geometry
• Poisson Quasi-Nijenhuis Manifolds
• Generalized complex structures and Lie brackets

Cited by in corpus (20)

• Reduction of Courant algebroids and generalized complex structures
• Symmetries in generalized Kähler geometry
• Hamiltonian symmetries and reduction in generalized geometry
• On higher analogues of Courant algebroids
• The Hitchin Model, Poisson-quasi-Nijenhuis Geometry and Symmetry Reduction
• E-Courant algebroids
• Generalized geometry, equivariant $\bar{\partial }\partial$-lemma, and torus actions
• Topology of generalized complex quotients
• Glanon groupoids
• Wick Rotations in Deformation Quantization
• The Equivariant cohomology theory of twisted generalized complex manifolds
• Dirac structures, nonholonomic systems and reduction
• Dirac geometry, quasi-Poisson actions and D/G-valued moment maps
• Complex Dirac structures: invariants and local structure
• VB-Courant algebroids, E-Courant algebroids and generalized geometry
• Induced Dirac structure on isotropy type manifolds
• Singular Reduction of Generalized Complex Manifolds
• Reduction and construction of Poisson quasi-Nijenhuis manifolds with background
• Generalized complex hamiltonian torus actions: Examples and constraints
• Reduction of $L_{\infty }$-Algebras of Observables on Multisymplectic Manifolds