Strong generalized holomorphic principal bundles

TL;DR

This paper introduces strong generalized holomorphic fiber bundles, developing their connection, curvature, and cohomology theories on generalized complex manifolds, with several examples and new theoretical insights.

Contribution

It defines SGH fiber bundles and develops their connection, curvature, and cohomology theories, extending classical concepts to the generalized complex setting.

Findings

Established de Rham and Dolbeault cohomologies for SGH bundles

Developed a Chern-Weil theory for SGH principal bundles

Presented Hodge theory and vanishing theorems for SGH vector bundles

Abstract

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.