Hamiltonian Symmetries and Reduction in Generalized Geometry

Given a close 3-form H ∈ Ω 0 (M), we define a twisted bracket on the space Γ(TM) ⊕ Ω 0 (M). We define the group of H-twisted Hamiltonian symmetries Ham(M, J;H) as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold, which leads to generalized complex reduction much the same way as symplectic reduction is constructed. The definitions and constructions are natural extensions of the corresponding ones in symplectic geometry. We describe cutting in generalized complex geometry to show that a general phenomenon in generalized geometry is that topology change is often accompanied by twisting (class) change.