# References for “modern” proof of Newlander-Nirenberg Theorem

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.

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Have you looked at the paper by Nijenhuis and Wolf in the Annals in the early sixties where they also prove existence of pseudoholomorphic curves. –  Mohan Ramachandran Mar 2 '11 at 21:23

There is a proof due to Malgrange which can be found in Nirenberg's, Lectures on Linear Partial Differential Equations. I am not sure that one can call the proof modern, but it is the simplest proof that I know.

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Thanks for this one too. I will check it out. –  Spiro Karigiannis Jul 7 '10 at 13:42

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

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Mister Hubbard, welcome to MO! –  Victor Protsak Mar 2 '11 at 21:05
@John: Something using the Banach space implicit function theorem and a split short exact sequence definitely sounds like a modern proof to me. I would certainly love to see the details of this. I'll contact you. Thanks! –  Spiro Karigiannis Mar 4 '11 at 12:31

It's covered in Demailly's too-little-known book, Complex analytic and differential geometry, though the proof given there is apparently modelled on the references you cited.

Edit: I just noticed that the MathOnline link currently seems to be non-functional, so here's a link to Demailly's webpage.

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Thanks. Indeed, I did not know about this book. It has a ton of useful other stuff too. I'd still be interested to hear about any other references, though. –  Spiro Karigiannis Jun 17 '10 at 15:39
+1 for the mathonline plug! :) –  Kevin H. Lin Jun 17 '10 at 17:32