Generalized complex structure on certain principal torus bundles

Abstract

A principal torus bundle over a complex manifold with even dimensional fiber and characteristic class of type $(1,1)$ admits a family of generalized complex structures. We show that such a generalized complex structure is equivalent to the product of the complex structure on the base and the symplectic structure on the fiber in a tubular neighborhood of the fiber. This has consequences for the generalized Dolbeault cohomology of the bundle.Comment: 19 page