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Local structure of generalized complex manifolds

Mohammed Abouzaid·Mitya Boyarchenko
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Abstract

We study generalized complex manifolds from the point of view of symplectic and Poisson geometry. We start by showing that every generalized complex manifold admits a canonical Poisson structure. We use this fact, together with Weinstein's classical result on the local normal form of Poisson manifolds, to prove a local structure theorem for generalized complex manifolds which extends the result Gualtieri has obtained in the "regular" case. Finally, we begin a study of the local structure of a generalized complex 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 generalized complex structure is encoded in the data of a constant B-field and a complex Lie algebra.

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Mohammed Abouzaid, Mitya Boyarchenko. 2004-12-04. Local structure of generalized complex manifolds. https://arxiv.org/abs/math/0412084

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