Local holomorphicity of generalized complex structures

Abouzaid and Boyarchenko showed that near any point of a generalized complex manifold there is a local model constructed as the product of a symplectic manifold with a generalized complex manifold which is of complex type at the point (similar to Weinstein's result on the local normal form of a Poisson structure). The question that remains of the local structure, then, is: what do generalized complex structures look like near a point of complex type? We show that they arise as a deformation of a (non-unique) complex structure by a holomorphic Poisson structure. We use a Nash-Moser type argument on shrinking neighbourhoods, in the style of Conn, adapting a more recent formalism of "scaled $C^\infty$ spaces" from Miranda, Monnier and Zung