Generalized geometric structures on complex and symplectic manifolds

Abstract

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 $$\left( M,j\right) $$ , 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 Kähler structures, or between hypercomplex and $$\mathbb {C}$$ -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 $$\left( M,\omega \right) $$ , each of them interpolating between two structures compatible with $$\omega $$ , for instance, between a $$\mathbb {C}$$ -symplectic and a para-Kähler structure (aka bi-Lagrangian foliation).