Reduction of generalized complex structures

We study reduction of generalized complex structures. More precisely, we investigate the following question. Let JJ be a generalized complex structure on a manifold MM, which admits an action of a Lie group GG preserving JJ. Assume that M0M0 is a GG-invariant smooth submanifold and the GG-action on M0M0 is proper and free so that MG≔M0/GMG≔M0/G is a smooth manifold. Under what condition does JJ descend to a generalized complex structure on MGMG? 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.