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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 10th 2023

    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 EME\to M together with a lifting of its R\mathbf{R}-module structure REnd(E)\mathbf{R}\to End(E) to a homomorphism of R\mathbf{R}-algebras CEnd(E)\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.

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