4 research outputs found
A neighbourhood theorem for submanifolds in generalized complex geometry
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.Comment: 36 pages, minor change
Blow-ups in generalized complex geometry
We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisson submanifolds and generalized Poisson transversals, submanifolds which look complex, respectively symplectic in transverse directions. We show that generalized Poisson submanifolds carry a canonical holomorphic ideal and give a necessary and sufficient condition for the corresponding blow-up to be generalized complex. For the generalized Poisson transversals we give a normal form for a neighborhood of the submanifold, and use that to define a generalized complex blow-up, which is up to deformation independent of choices
Blow-ups in generalized complex geometry
We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisson submanifolds and generalized Poisson transversals, submanifolds which look complex, respectively symplectic in transverse directions. We show that generalized Poisson submanifolds carry a canonical holomorphic ideal and give a necessary and sufficient condition for the corresponding blow-up to be generalized complex. For the generalized Poisson transversals we give a normal form for a neighborhood of the submanifold, and use that to define a generalized complex blow-up, which is up to deformation independent of choices