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Is there an example of a complex bundle on $\mathbb CP^n$ or on a Fano variety (defined over complex numbers), that does not admit a holomorphic structure? We require that the Chern classes of the bundle are $(k,k)$ Hodge classes (which is automatic for $\mathbb CP^n$ or Fanos of dimension<4). If by any chance such examples are known, what is the smallest dimension of the variety (or the bundle)?

For $\mathbb CP^1$ it is elementary to see that all bundles are holomorphic. In the book of Okonek and Schneider it is stated, that all complex bundles on $\mathbb CP^2$ and $\mathbb CP^3$ are also holomorphic. But for $\mathbb CP^n$, $n\ge 4$ this is stated as an open problem (as for 1980).

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Here is the answer to the question, kindly explained to me by Burt Totaro.

EDITED. This is an OPEN PROBLEM.

0) Apparently in the case of $\mathbb CP^n$ existence of a complex bundle without holomorphic structure is still an OPEN PROBLEM. Though it is believed that there should be plenty of examples starting from $n\ge 5$, coming from topologically indecomposable rank two bundles, apparently no such bundle was proven to be non-holomorphic as for today.

1) A topologically non-trivial rank 2 complex bundle with $c_1=0$, $c_2=0$ was constructed in

Rees, Elmer, Some rank two bundles on ${\rm P}_{n}\mathbb C$, whose Chern classes vanish. Variétés analytiques compactes (Colloq., Nice, 1977).

It was also claimed in this article that this bundle does not admit a holomorphic structure. But this claim was deduced from an article that contained a gap. So it is yet unknown if this particular bundle has holomorphic structure or not. This is discussed in M. Schneider. Holomorphic vector bundles on ${\rm P}^n$. Seminaire Bourbaki 1978/79, expose 530.

This is why Okonek and Schneider write in their book p. 137 that this is an open problem.

2) On the positive side it is proven that every complex vector bundle on a smooth projective rational 3-fold has an holomorphic structure.

C. Banica and M. Putinar. On complex vector bundles on projective threefolds. Invent. Math. 88 (1987), 427-438.

3) If one wants to construct examples of bundles on projective manifolds that are not necessarily Fanos it is possible to use the fact that the integral Hodge conjecture fails. Namely there are elements in $H^{2p}(X,\mathbb Z)$ which are in $H^{p,p}$ but which are not represented by an algebraic cycle. Kollar gave such examples with $dim(X)=3$. A recent reference, which refers back to earlier results, is:

C. Soule and C. Voisin. Torsion cohomology classes and algebraic cycles on complex projective manifolds. Adv. Math. 198 (2005), 107-127

4) One reason to expect examples of such bundles in higher dimensions is Schwarzenberger's conjecture that every rank-2 algebraic vector bundle $E$, on $\mathbb CP^n$ with $n\ge 5$ is a direct sum of two line bundles. So, for example, if $c_1(E)=0$ then $c_2(E)=-d^2$ for some integer $d$, according to the conjecture.

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    $\begingroup$ 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. $\endgroup$ – Greg Kuperberg Dec 2 '09 at 20:55
  • $\begingroup$ 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? $\endgroup$ – algori Dec 3 '09 at 0:30
  • $\begingroup$ 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 :) $\endgroup$ – Dmitri Panov Dec 4 '09 at 10:31
  • $\begingroup$ 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... $\endgroup$ – Dmitri Panov Dec 4 '09 at 19:33

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