TITLE: References for "modern" proof of Newlander-Nirenberg Theorem
QUESTION [24 upvotes]: Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In particular, I want to concentrate on the Hodge theorem, the Newlander-Nirenberg theorem, and the Calabi-Yau theorem.
I have many excellent references (and have lectured before) on the Hodge and CY theorems. However, for the Newlander-Nirenberg theorem, I am finding it hard to find a "modern" treatment. I recall going through the original paper in my graduate student days, but I hope that there is a more streamlined version floating around somewhere. (I want to consider the general smooth case, not the easy real-analytic version). Besides the original paper, so far I can only find these references:
J. J. Kohn, "Harmonic Integrals on Strongly Pseudo-Convex Manifolds, I and II" (Annals of Math, 1963)
and
L. Hormander, "An introduction to complex analysis in several variables" (Third Edition, 1990)
Both are easier to follow than the original paper. But my question is: are there any other proofs in the literature, preferably from books rather than papers? The standard texts on complex and Kahler geometry that I have looked at don't have this.

REPLY [21 votes]: Hi Spiro:  I have had much the same difficulties as you, but I now know a modern proof.
At heart, the original proof is an application of the implicit function theorem. More specifically, let $U$ be a polydisk in $C^n$ consider the sequence of Banach manifolds
$Diff^{k,\alpha}(U,C^n) \to AC^{k-1,\alpha}(U) \to (A^{0,2})^{k-2,\alpha}(U,TU).$
These are respectively the diffeomorphisms $U\to C^n$ of class $(k,\alpha)$, the almost complex structures on $U$ of class $(k-1,\alpha)$ and the $(0,2)$ forms on $U$ with values in the holomorphic tangent bundle, of class $(k-2,\alpha)$.  The first map is the pullback of the standard complex structure, and the second is the Frobenius integrability form 
$\phi \mapsto \overline \partial \phi - \frac 12 [\phi\wedge \phi].$ 
The object is to show that the first map is locally surjective onto the inverse image of $0$ by the second. These spaces are Banach manifolds, and if you can show that the sequence  of derivatives (respectively at the identity, at the standard complex structure and at 0) is split exact, the result follows from the implicit function theorem.
This sequence of derivatives is the Dolbeault sequence on $U$ (in the appropriate class), and it is split exact, though this is NOT obvious.  There is an error in the original paper, or rather in the paper of Chern's that it depends on, but the result is true.  The remainder of the mess in the original proof is due to the authors writing out the Picard iteration in the specific case, rather than isolating the needed result.
I am working on getting this written up with Milena Pabiniak, a graduate student here at Cornell.  Write me at jhh8@cornell.edu  if you are interested in seeing details.
John Hubbard