Confusion surrounding the Koszul-Malgrange theorem

Question

I recently had the need to appeal to some complex geometry in my research and have been trying to unravel the various relationships surrounding the Koszul-Malgrange theorem.

According to nlab, the theorem goes as follows.

Theorem 1. (Koszul-Malgrange theorem)

Holomorphic vector bundles over a complex manifold are equivalently complex vector bundles which are equipped with a (hermitean) holomorphic flat connection. Under this identification the Dolbeault operator $\overline{\partial} $ acting on the sections of the holomorphic vector bundle is identified with the holomorphic component of the covariant derivative of the given connection.

Since the original reference is in french, which I have trouble reading, I haven't been able to go though the proof of the theorem. From the bit of complex geometry I know though, I am a bit confused.

I know that, given a complex vector bundle $E\to X$ and a flat connection, the $(0,1)$-part of the connection defines a holomorphic structure, while the $(1,0)$-component defines a holomorphic connection on the corresponding holomorphic bundle. By Chern-Weil theory, the rational chern classes of $E$ vanish, as do the rational Chern classes of a holomorphic vector bundle equipped with holomorphic connection.

On the other hand, I know that in general, a holomorphic bundle may have nonvanishing rational Chern classes. In fact, over compact, complex projective manifolds, these classes generate certain Hodge classes (or possibly all according to the Hodge conjecture).

This makes me think the theorem should be a correspondence between flat bundles and holomorphic vector bundles with holomorphic connection.

Question

What is the exact relationship between smooth complex vector bundles equipped with connection and holomorphic structures? I realize this question is probably a bit elementary, but complex geometry is not my specialty. References are also welcome.

Answer 1

Since I was rather surprised that those authors would make such a claim, I looked up the original reference [Koszul, Malgrange, Sur certaines structures fibrées complexes.] Here is a rough translation of the relevant theorem:

Theorem 2. Let $G$ be a complex Lie group, $V$ a complex manifold, $P$ a principal bundle with group $G$ over $V$. Let $\gamma$ be a connection form on $P$. The following are equivalent

• The almost complex structure defined by $\gamma$ is a complex structure.
• ...
• The $(0,2)$ part of the curvature $R_\gamma$ is $0$.
• ...

[Footnote in orignal: ...For $R_\gamma$ to be of type $(2,0)$ it is necessary and sufficient that $\gamma$ be holomorphic...]

So that they are not saying that the curvature is zero, or that the connection is holomorphic. Clearly the version of the theorem you quoted is needlessly ambiguous, at best. Perhaps you should contact the author of that nlab page.