  We study neighbourhoods of submanifolds in generalized complex geometry. Our
first main result provides sufficient criteria for such a submanifold to admit
a neighbourhood on which the generalized complex structure is B-field
equivalent to a holomorphic Poisson structure. This is intimately tied with our
second main result, which is a rigidity theorem for generalized complex
deformations of holomorphic Poisson structures. Specifically, on a compact
manifold with boundary we provide explicit conditions under which any
generalized complex perturbation of a holomorphic Poisson structure is B-field
equivalent to another holomorphic Poisson structure. The proofs of these
results require two analytical tools: Hodge decompositions on almost complex
manifolds with boundary, and the Nash-Moser algorithm. As a concrete
application of these results, we show that on a four-dimensional generalized
complex submanifold which is generically symplectic, a neighbourhood of the
entire complex locus is B-field equivalent to a holomorphic Poisson structure.
Furthermore, we use the neighbourhood theorem to develop the theory of blowing
down submanifolds in generalized complex geometry.
