Blow-ups in generalized complex geometry

We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideals, 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. These are submanifolds for which the geometry normal to the submanifold is complex, respectively symplectic. We show that generalized Poisson submanifolds carry a canonical holomorphic ideal, and we give a necessary and sufficient condition for the corresponding blow-up to be generalized complex. For generalized Poisson transversals we prove a normal form theorem for a neighborhood of the submanifold and use it to define a generalized complex blow-up.