# Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

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

• Aren't complex tori Oka manifolds? Aug 31 '15 at 9:33
• Oops, complex tori don't work: $h^{0,1}$ is always nonzero. Aug 31 '15 at 13:39
• 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..? May 18 '17 at 10:51
• @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? May 25 '17 at 9:52
• yeah, that is true. I don't know how to rule that out. May 25 '17 at 12:14

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

• 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 $D$.
– user21574
Jul 31 '17 at 0:12