I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up.


Let $X$ be a complex manifold. Let $E$ be a smooth, complex vector bundle. Let $C^{\infty}(E)$ be the sheaf of smooth sections of $E$. A dee-bar connection is a $\mathbb{C}$-linear map $\nabla : C^{\infty}(E) \to C^{\infty}(E) \otimes \Omega^{0,1}$, obeying the Liebnitz rule
$$\nabla(f \sigma) = f \nabla(\sigma) + \bar{\partial}f \otimes \sigma$$
Just as with connection in the $C^{\infty}$ setting, we can define the curvature; it is a section of $\Omega^{0,2} \otimes \mathrm{End}(E)$. Suppose that the curvature is zero.


Let $\mathrm{Ker}(\nabla)$ be the kernel of $\nabla : C^{\infty}(E) \to C^{\infty}(E) \otimes \Omega^{0,1}$. Let $\mathcal{O}$ be the sheaf of homolomorphic functions. It is easy to see that $\mathrm{Ker}(\nabla)$ is a $\mathcal{O}$-module. I want a reference for the fact that:



> 
> $\mathrm{Ker}(\nabla)$ is a locally free $\mathcal{O}$-module of rank $\dim E$.
> 
> 
> 


The analogous fact for smooth connections is (one direction of) the Riemann-Hilbert correspondence; see, for example, [Theorem 2.6 here](http://math.stanford.edu/~conrad/papers/rhtalk.pdf).



Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle
This is Theorem 2.1.53 in Donaldson--Kronheimer's *Geometry of 4-manifolds*. They prove it from scratch via the contraction mapping principle but there are arguments from the Newlander--Nirenberg theorem also.


