  A closed 3-form $H \in \Omega^3_0(M)$ defines an extension of $\Gamma(TM)$ by
$\Omega^2_0(M)$. This fact leads to the definition of the group of $H$-twisted
Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie
group and moment map in the category of (twisted) generalized complex manifold.
The Hamiltonian reduction in the category of generalized complex geometry is
then constructed. The definitions and constructions are natural extensions of
the corresponding ones in the symplectic geometry. We describe cutting in
generalized complex geometry to show that it's a general phenomenon in
generalized geometry that topology change is often accompanied by twisting
(class) change.
