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nLab > Latest Changes: projective bundle

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2017

    created a minimum at projective bundle

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 29th 2017

    I can add some observation about how not every projective bundle is the projectivisation of a vector bundle, etc. in the coming days.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2017

    Thanks. When you do that, you should also state the definition (I suppose fiber bundle with a projective space as the typical fiber) because at the moment the entry only talks about projectivisation of vector bundles.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 29th 2017

    Will do.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2020
    • (edited Dec 30th 2020)

    added a basic note (here) on the splitting off of line bundles in the pullback bundle to a projective bundle

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 30th 2020

    I think also the complete splitting could be executed in one shot by pulling back to a “flag bundle” associated to a vector bundle. I’m not sure what terminology people use though.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2020

    Sure, as in the reference given. Discussion of that might naturally go to a page flag bundle.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 31st 2020

    spelled out (here) the example of complex projective bundles of tautological quaternionic line bundles:

    P ℂ(ℒ ℍP n) ≃ ℂP 2n+1 ↘ ↙ v⋅ℂ ×↦v⋅ℍ × ℍP n \array{ P_{\mathbb{C}} \big( \mathcal{L}_{{}_{\mathbb{H}P^n}} \big) && \simeq && \mathbb{C}P^{2n+1} \\ & \searrow && \swarrow_{ \mathrlap{ v \cdot \mathbb{C}^\times \,\mapsto\, v \cdot \mathbb{H}^\times } } \\ && \mathbb{H}P^n }

    diff, v5, current

1 to 8 of 8

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