Journal of Symplectic Geometry
Volume 4 (2006)
Local structure of generalized complex manifolds
Pages: 43 – 62
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.