  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 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.
