Blow-ups in generalized complex geometry

TL;DR

This paper develops a framework for blow-ups in generalized complex geometry by introducing holomorphic ideals and analyzing specific submanifolds, expanding the understanding of geometric transformations in this field.

Contribution

It introduces the concept of holomorphic ideals for blow-ups and characterizes when generalized complex blow-ups are possible, focusing on generalized Poisson submanifolds and transversals.

Findings

Holomorphic ideals enable blow-ups in smooth manifolds.

Characterization of when blow-ups preserve generalized complex structures.

Normal form for neighborhoods of generalized Poisson transversals.

Abstract

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.