Does every compact complex manifold of complex dimension greater than or equal two possess a nontrivial rank 2 holomorphic vector bundle?

  • $\begingroup$ Aren't complex tori Oka manifolds? $\endgroup$ Aug 31 '15 at 9:33
  • $\begingroup$ Oops, complex tori don't work: $h^{0,1}$ is always nonzero. $\endgroup$ Aug 31 '15 at 13:39
  • $\begingroup$ Consider the Calabi-Eckman manifold = $S^3 \times S^3$, I would guess that since $b_{2} = b_{4} = 0$, that there is not even non-trivial rank 2 complex bundle..? $\endgroup$
    – Nick L
    May 18 '17 at 10:51
  • $\begingroup$ @Nick Lindsay But we may still have some nontrivial holomorphic structures in the trivial complex vector bundle over the Calabi-Eckman manifold you have mentioned. Am I right? $\endgroup$
    – Hamed
    May 25 '17 at 9:52
  • $\begingroup$ yeah, that is true. I don't know how to rule that out. $\endgroup$
    – Nick L
    May 25 '17 at 12:14

Some partial answer:

On non-Kahler surfaces we can give stronger version of your question and we have allways the following thorem

Theorem: Let $S$ be non-Kahler surfaces, For any $n > 0$ there exists a Mumford stable rank-2 vector bundle $E$ with trivial determinant, ($det(E) = \mathcal O_S$ ) and $c_2(E)=n$.

We have also the following theorem due to Banica - Le Potier

Theorem: If $S$ is a non-projective surface and $N$ is enough big , then there exists rank-2 vector bundles on $S$ with $c_2(E) = N$ which are non-filtrable.

We have following theorem due to Ballico and Schwarzenberger

Theorem: If $S$ be a projective surface and $E$ is any topological rank-2 vector bundle with $det(E) \in Pic(X)$, then $E$ has a holomorphic structure by applying elementary transformations starting from the trivial bundle.

We have Cartan-Serre theorem for construction of rank 2 vector bundles on projetive varieties.

Theorem: Let $X$ be a complex manifold, $L_1, L_2 ∈ Pic(X)$ are line bundles, $Z ⊂ X$ with $codim_X(Z) = 2$. Under some cohomological conditions, the sheaf $E$ sitting in

$$0 \to L_1 \to E \to L_2 \otimes J_Z → o$$ is locally free. Moreover, If $X$ is projective, any rank-2 holomorphic vector bundle can be constructed this way

  • 4
    $\begingroup$ Additional comment: Let $D$ be a smooth irreducible divisor on a smooth variety $X$. Then an element $\mu\in H^1(D,\mathcal N_{D/X}) $ defines a rank 2 vector bundle over $ D$. $\endgroup$
    – user21574
    Jul 31 '17 at 0:12

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