Local structure of generalized complex manifolds

Abstract

We study generalized complex (GC) manifolds from the point of view of symplectic and Poisson geometry. We start by recalling that every GC manifold admits a canonical Poisson structure. We use this fact, together with Weinstein's classical result on the local normal form of Poisson, to prove a local structure theorem for GC, complex manifolds, which extends the result Gualtieri has obtained in the "regular'' case. Finally, we begin a study of the local structure of a GC manifold in a neighborhood of a point where the associated Poisson tensor vanishes. In particular, we show that in such a neighborhood, a "first-order approximation'' to the GC structure is encoded in the data of a constant B-field and a complex Lie algebra.