Timeline for $\mathbb CP^k$ bundles over $\mathbb CP^n$ are projectivisations of vector bundles. Any reference?
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11 events
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| Dec 2, 2019 at 19:54 | comment | added | aglearner | Daniel, I would be very happy to see such an answer. Even if not a reference, I think this would benefit the mathematical community (maybe you have noticed from the comments above, even those who red Hartschorne got a bit puzzled by the question). So I am determined for this question to get a proper answer on this website :) | |
| Dec 2, 2019 at 18:50 | comment | added | Daniel Litt | @aglearner: If you so desire, you can take the proof Angelo has sketched and rewrite it purely in terms of Cech cocycles. This is as elementary a proof as I can imagine... | |
| Dec 1, 2019 at 17:34 | comment | added | aglearner | Denis, you are right of course. But yet, could you imagine that I am the first mathematician in this world who wrote down this statement? Am I the first one who needed it? If yes, I would be quite astonished... | |
| Dec 1, 2019 at 17:30 | comment | added | Denis Nardin | @aglearner I'm not sure trying to avoid the methods of "modern" algebraic geometry will be fruitful (Angelo's proof uses only sheaf cohomology, and I'm pretty sure even complex analytic algebraic geometry needs that). | |
| Dec 1, 2019 at 13:53 | comment | added | aglearner | Dear Angelo, so do you think that one could find a proof of this statement or reference to it without appealing to Grothendieck? I tried for a few months, but failed. Hence my question (none of algebraic geometers whom I asked in real life gave me a reference... - only proofs via Brauer group) | |
| Dec 1, 2019 at 12:47 | comment | added | Angelo | You are trying to lift a holomorphic $\mathrm{PGL}_{k+1}$-principal bundle to a $\mathrm{GL}_{k+1}$-principal bundle. Using standard methods in bundle theory, this reduces to the statement that $\mathrm H^2(\mathbb P^n, \mathcal O^*) = 0$, where $\mathcal O^*$ is the sheaf of invertible holomorphic bundle. From the exponential sequence this reduces to the statements that $\mathrm H^2(\mathbb P^n, \mathcal O) = \mathrm H^3(\mathbb P^n, \mathbb Z) = 0$, which are standard. | |
| Nov 30, 2019 at 20:43 | comment | added | aglearner | Dear Arun, I see. I realised that I have not stated in my question that I need a reference for the fact that these two spaces are isomorphic as $\it projective\; varieties$. So, this is not quite a question in algebraic topology, it is rather a question in algebraic geometry. It is a statement that there is an invertible holomorphic map between two spaces. I'll update the question so that this is stated explicitly. | |
| Nov 30, 2019 at 19:26 | comment | added | Arun Debray | @aglearner I had not seen this statement before, though this kind of argument is related to things I spend a lot of time thinking about. | |
| Nov 30, 2019 at 19:24 | comment | added | Arun Debray | @MikeMiller yes, I think so. It might be possible to upgrade it to a holomorphic equivalence using holomorphic charts on $\mathbb{CP}^n$ or something, but I can't think of how. | |
| Nov 30, 2019 at 19:00 | comment | added | mme | This only shows the resulting statement up to diffeomorphism, right? | |
| Nov 30, 2019 at 18:54 | history | answered | Arun Debray | CC BY-SA 4.0 |