It is a well-known fact that the partitions of unity are used quite often when dealing with the proof of some technical real-manifold theorem (e.g. Stokes). The advantage of this kind of arguments is that you can reduce your proof to an open subset of the manifold that is dipheomorphic to some $\mathbb{R}^n$ and, in the last instance, you can reduce it to $\mathbb{R}^n$.
My question is "What happens when you are working with a complex manifold and its holomorphic charts?" Is there some mechanism (maybe not so good as partitions of unity) that can "enlighten our job" or it is just about brute strength treatment?
It would be helpful some link or reference to a bibliographical item where some technical proof (it does not matter the theorem dealt) for a complex manifold is developed.