# A neighbourhood theorem for submanifolds in generalized complex geometry

## TL;DR

This paper establishes criteria for neighborhoods of submanifolds in generalized complex geometry to resemble holomorphic Poisson structures and proves a rigidity theorem for their deformations, with applications to blow-down procedures.

## Contribution

It provides new conditions for neighborhoods to be B-field equivalent to holomorphic Poisson structures and introduces a rigidity result for deformations on compact manifolds with boundary.

## Key findings

- Neighborhoods of certain submanifolds are B-field equivalent to holomorphic Poisson structures.
- Rigidity of generalized complex deformations of holomorphic Poisson structures is established.
- Application to blow-down operations in generalized complex geometry.

## Abstract

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.