ON COTANGENT MANIFOLDS, COMPLEX STRUCTURES AND GENERALIZED GEOMETRY

We develop various properties of symmetric generalized complex structures (in connection with their holomorphic space and B-field transformations), which are analogous to the well-known results of Gualtieri on skew symmetric generalized complex structures. Given an adapted (symmetric or skew symmetric) generalized complex structure J and a linear connection D on a manifold M, we construct an almost complex structure J(J,D) on the cotangent manifold T* M and we study its integrability. For J skew-symmetric, we relate the Courant integrability of J with the integrability of JJ,D. We consider in detail the case when M = G is a Lie group and J, D are left-invariant. We also show that our approach unifies and generalizes various known results from special complex geometry.