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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 10th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## Definition A _complex vector bundle_ is a [[vector bundle]] whose fibers are [[complex vector spaces]]. A complex vector bundle with complex 1-dimensional [[fibers]] is a [[complex line bundle]]. More precisely, a __complex vector bundle__ is a [[real vector bundle]] $E\to M$ together with a lifting of its $\mathbf{R}$-module structure $\mathbf{R}\to End(E)$ to a homomorphism of $\mathbf{R}$-algebras $\mathbf{C}\to End(E)$. The latter lifting is also known as a __complex structure__ on a [[real vector bundle]]. In terms of [[cocycles]], complex vector bundles can be described using cocycle data where the transition maps are complex-linear maps. ## Connections to holomorphic vector bundles Any [[holomorphic vector bundle]] over a [[complex manifold]] has an underlying complex vector bundle. Conversely, given a complex vector bundle over a [[complex manifold]], if its transition maps are holomorphic, then it is a [[holomorphic vector bundle]]. <a href="https://ncatlab.org/nlab/revision/diff/complex+vector+bundle/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/complex+vector+bundle/12">v12</a>, <a href="https://ncatlab.org/nlab/show/complex+vector+bundle">current</a>

   Added:

   Definition

   A complex vector bundle is a vector bundle whose fibers are complex vector spaces.

   A complex vector bundle with complex 1-dimensional fibers is a complex line bundle.

   More precisely, a complex vector bundle is a real vector bundle $E\to M$ together with a lifting of its $\mathbf{R}$-module structure $\mathbf{R}\to End(E)$ to a homomorphism of $\mathbf{R}$-algebras $\mathbf{C}\to End(E)$.

   The latter lifting is also known as a complex structure on a real vector bundle.

   In terms of cocycles, complex vector bundles can be described using cocycle data where the transition maps are complex-linear maps.

   Connections to holomorphic vector bundles

   Any holomorphic vector bundle over a complex manifold has an underlying complex vector bundle.

   Conversely, given a complex vector bundle over a complex manifold, if its transition maps are holomorphic, then it is a holomorphic vector bundle.

   diff, v12, current