Timeline for answer to Complex vector bundles that are not holomorphic by Dmitri Panov
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| Dec 1, 2018 at 21:46 | history | edited | Dmitri Panov | CC BY-SA 4.0 |
typos corrected
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| Dec 5, 2009 at 1:17 | vote | accept | Dmitri Panov | ||
| Dec 4, 2009 at 19:33 | comment | added | Dmitri Panov | PS. Since this problem is open, I deduce that non-trivial holomorphic bundles with $c_1=0=c_2$ exist on $CP^2$ :). Overwise it would be possible to prove that Rees bundle does not have holomorphic structure... | |
| Dec 4, 2009 at 14:20 | history | edited | Dmitri Panov | CC BY-SA 2.5 |
added 785 characters in body
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| Dec 4, 2009 at 10:31 | comment | added | Dmitri Panov | Algori, I hope the answer to your question is in Rees paper, but I could not get hold of it yet. As far as I understood if you consider a rank 2 holomorphic bundle over $CP^n$ with $c_1=0=c_2$ it will be always trivial starting from some $n$. But I was not able to prove it, or to see what happen, even for for $CP^2$. One idea is to assume by contradiciton that such bundle exists, then it will be unstable. In the case of $CP^2$ this would mean that the bundle contains a sub-line budle $O(k)$, $k\ge 1$. But this did not help me to find a proof yet :) | |
| Dec 3, 2009 at 0:30 | comment | added | algori | Dmitri -- thanks for asking this question, and also for answering it! Regarding 1) above: how does one prove that the rank 2 bundle on $mathbb{P}^5(\mathbb{C})$ with vanishing Chern classes you mention does not admit a holomorphic structure? | |
| Dec 2, 2009 at 20:55 | comment | added | Greg Kuperberg | I'll just briefly add that you can start to see Rees' construction coming once you calculate that $SU(2) = S^3$ has many interesting higher homotopy groups. It means that $BSU(2)$ does too, so that you eventually expect many interesting plane bundles with vanishing Chern classes. (Provided that the Postnikov towers allow it, which they happen not to when $n=4$.) I did not know about any of this great work, but I noticed that much. | |
| Dec 2, 2009 at 17:39 | history | answered | Dmitri Panov | CC BY-SA 2.5 |