Generalized geometric structures on complex and symplectic manifolds

Abstract

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