On the local and global classification of generalized complex structures

TL;DR

This paper advances the understanding of generalized complex structures by providing local and global classification results, characterizing their local form, cohomological criteria, and structures with nondegenerate type change.

Contribution

It offers new classification theorems, cohomological criteria, and explicit examples, deepening the understanding of generalized complex geometry.

Findings

Generalized complex structures near complex points derive from holomorphic Poisson structures.

Counterexamples show not all regular Poisson and transverse complex structures come from generalized complex structures.

A local normal form theorem describes structures with nondegenerate type change.

Abstract

We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a smoothed Newton's method along the lines of Nash, Moser and Conn. In the second topic, we consider whether a given regular Poisson structure and transverse complex structure come from a generalized complex structure. We give cohomological criteria, and we find some counterexamples and some unexpected examples, including a compact, regular generalized complex manifold for which nearby symplectic leaves are not symplectomorphic. In the third topic, we consider generalized complex structures with nondegenerate type change; we describe a generalized Calabi-Yau structure induced on the type change locus, and prove a local normal form theorem near this locus. Finally, in the fourth topic, we give a classification of generalized complex principal bundles satisfying a certain transversality condition; in this case, there is a generalized flat connection, and the classification involves a monodromy map to the Courant automorphism group.