Local structure of generalized complex manifolds

TL;DR

This paper investigates the local geometric structure of generalized complex manifolds by linking them to Poisson geometry, extending known results, and analyzing neighborhoods where the Poisson tensor vanishes.

Contribution

It establishes a local structure theorem for generalized complex manifolds using Poisson normal forms and explores neighborhoods with vanishing Poisson tensors.

Findings

Every generalized complex manifold admits a canonical Poisson structure.

A local normal form theorem extends Gualtieri's regular case result.

Neighborhoods where the Poisson tensor vanishes are characterized by a B-field and a complex Lie algebra.

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.