Generalized geometric structures on complex and symplectic manifolds

On a smooth manifold M, generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M. Given a complex manifold (M, j), we define six families of distinguished generalized complex or paracomplex structures on M. Each one of them interpolates between two geometric structures on M compatible with j, for instance, between totally real foliations and Kahler structures, or between hypercomplex and C-symplectic structures. These structures on M are sections of fiber bundles over M with typical fiber G/H for some Lie groups G and H. We determine G and H in each case. We proceed similarly for symplectic manifolds. We define six families of generalized structures on (M, omega), each of them interpolating between two structures compatible with., for instance, between a C-symplectic and a para-Kahler structure (aka bi-Lagrangian foliation).