2012/01/04 by Michael Bailey, Bailey, Michael · 1 citation
Mathematics · #53D18 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1201.0791
openalex publication_date 2012/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.