Principal Bundles in Generalized Complex Geometry

Abstract

In this thesis, we introduce the notion of a strong generalized holomorphic (SGH) fiber bundle and develop connection and curvature theory for an SGH principal G-bundle over a regular generalized complex (GC) manifold, where G is a complex Lie group. We also develop a de Rham cohomology for regular GC manifolds, and a Dolbeault cohomology for SGH vector bundles. Moreover, we establish a Chern-Weil theory for SGH principal G-bundles under certain mild assumptions on the leaf space of the GC structure. We also present a Hodge theory along with associated dualities and vanishing theorems for SGH vector bundles. Several examples of SGH fiber bundles are given. Additionally, we describe a family of regular generalized complex structures (GCS) on a principal torus bundle over a complex manifold with even dimensional fiber and characteristic class of type (1, 1). The leaves of the associated symplectic foliation are exactly the fibers of the bundle. The speciality of such principal torus bundles is that they are not always SGH principal bundles. We show that such a GCS is equivalent to the product of the complex structure on the base and the symplectic structure on the fiber in a tubular neighborhood of an arbitrary fiber if and only if the bundle is flat, impacting the generalized Dolbeault cohomology of the bundle with a Künneth formula. Moreover, if a principal bundle over a complex manifold with a symplectic structure group admits a GCS with the fibers of the bundle as leaves of the associated symplectic foliation, and the GCS is equivalent to a product GCS in a neighborhood of every fiber, then we show that the bundle is flat and symplectic. This is similar to the behavior of SGH principal bundles over a complex manifold with symplectic fibers.