I am looking for a reference which shows that the following statements are equivalent for a complex vector bundle $E$:


* $E$ is a holomorphic vector bundle.
* There is a Dolbeault operator $\bar{\partial}\_E$, i.e. a $\mathbb{C}$ linear operator $\bar{\partial}\_E : \Omega^{0,0}(E) \to \Omega^{0,1}(E)$ which satisfies the Leibniz rule and $\bar{\partial}\_E^2 = 0$.


This is stated without proof in Huybrechts' *Complex Geometry: An Introduction*.



$E$ is a holomorphic vector bundle if and only if there is a Dolbeault operator $\bar{\partial}_E$
A. Moroianu gives a detailed proof on pp. 72-74 of his [*Lectures on Kähler geometry*](http://www.ams.org/mathscinet-getitem?mr=2325093) (Theorem 9.2), available on the internet. (The [preprint](http://arxiv.org/abs/math/0402223v1) has it as Theorem 3.2.)


He attributes that proof to S. Kobayashi, [*Differential geometry of complex vector bundles*](http://www.ams.org/mathscinet-getitem?mr=909698). I guess he means Proposition I.3.7 there, which Kobayashi couches in a different language involving connections.


**P.S.** Note that Moroianu gives himself a whole *complex* of operators $\bar\partial\_E:\Omega^{p,q}(E)\to\Omega^{p,q+1}(E)$, and Donaldson-Kronheimer at least the $(0,q)$ ones, plus Leibniz. I'm not 100% clear that just the $(0,0)$ one plus Leibniz suffice to determine everything, as the question seems to imply.


