Principal bundles on compact complex manifolds with trivial tangent bundle
Indranil Biswas

TL;DR
This paper characterizes when holomorphic principal bundles over compact quotients of complex Lie groups admit holomorphic or flat connections, linking invariance and homogeneity to the existence of such connections.
Contribution
It establishes necessary and sufficient conditions for the existence of holomorphic and flat holomorphic connections on principal bundles over certain complex manifolds.
Findings
Holomorphic principal bundles admit a holomorphic connection iff they are invariant.
In simply connected cases, flat holomorphic connections exist iff the bundle is homogeneous.
Provides a complete characterization of connections on bundles over quotients of complex Lie groups.
Abstract
Let be a connected complex Lie group and a cocompact lattice. Let be a complex Lie group. We prove that a holomorphic principal -bundle over admits a holomorphic connection if and only if is invariant. If is simply connected, we show that a holomorphic principal -bundle over admits a flat holomorphic connection if and only if is homogeneous.
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Taxonomy
TopicsGeometry and complex manifolds · Advanced Algebra and Geometry · Geometric Analysis and Curvature Flows
