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The Newlander-Nirenberg theorem February 27, 2009

Posted by Joel Kamnitzer in Algebraic Geometry, differential geometry, Paper Advertisement.
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As you may have noticed, I haven’t written a post in a long time.  Fortunately my fellow bloggers have been doing a great job in my absence.  Though I haven’t been posting, I have been reading and enjoying the blog regularly.

If you want to know what I have been up to in the meantime, the short answer is “Bella”. For a longer, more mathematical answer, you could check out this post or look at the recent papers that I’ve posted on the arxiv with Sabin Cautis and Tony Licata.

In this post, however, I would like to discuss something completely different, namely the Newlander-Nirenberg theorem. Let me begin by recalling the setting. An almost complex structure on a smooth manifold M is an endomorphism J of the tangent bundle such that J^2 = -1 .

One way to get an almost complex structure is to start with a complex manifold. The definition of a complex manifold is just like that of a smooth manifold — ie in terms of an atlas — except that we require that the transition functions be holomorphic.

The underlying real manifold of a complex manifold has an almost complex structure. An almost complex structure which arises in this way is called integrable. The N-N theorem gives you a criterion for testing whether an almost complex structure is integrable.

I am teaching a course in symplectic geometry this semester and I’m trying to do a fair bit of complex geometry, so I thought that I should cover this theorem. However, before the past week, I didn’t know much about the proof of this theorem. Now, that I’ve understood a few things, I would like to share them with you.

Recall from David’s post , the notion of a (0,1) vector field. Actually, David discussed (0,1) forms, but (0,1) vector fields are just the dual notion. Namely they are those complex valued vector fields M which are eigenvectors for J with eigenvalue -i .

The N-N theorem can be stated as follows:

Theorem
An almost complex structure (M, J) is integrable iff the (0,1) vector fields on M are closed under the Lie bracket.

This statement is reminiscent of the Frobenius theorem on the integrability of distributions on a real manifold and I always knew there was a relation but it was never clear to me what exactly the relation was. The basic question is: on which manifold do the (0,1) vector fields give a distribution?

After reading in Voisin’s excellent book “Hodge theory and complex algebraic geometry”, I know understand the answer to this question and I can give an outline of an (incomplete) proof for the Theorem.

Let me start by assuming that the manifold M is real analytic (this is source of the incompleteness).  This means that it has an atlas whose transition functions are all real analytic — ie given by power series.  Then I can complexify my manifold to obtain M_{\mathbb{C}} .  This is a complex manifold, whose complex dimension is the same as the real dimension of M.  It is given by an atlas whose transition functions are the same power series as those for M , except they are now viewed complex analytic functions.

Now the complexified tangent bundle of M is the actual tangent bundle of M_{\mathbb{C}} and so the subspace of (0,1) vector fields gives a holomorphic distribution on M_{\mathbb{C}} (here we also assume that the original complex structure J was real analytic).  Then we apply a holomorphic version of the Frobenius theorem to show that locally we can find a holomophic map U \rightarrow \mathbb{C}^n whose kernel is this distribution.  Then we show that the restriction of this holomorphic map to M \cap U \subset U is a diffeomorphism. By definition, this map will be J-holomorphic and thus we build an atlas on M. This completes the proof.  Pretty nifty, eh?

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Comments

1. Ben Wieland - March 2, 2009

The proof’s not incomplete; it’s just a proof of a weaker theorem! The weaker theorem may be all you need.

Whitney proved that all (compact?) smooth manifolds are canonically real analytic. Namely, embed it in euclidean space and express it as (a component of) the intersection of the vanishing loci of polynomials. It’s a real algebraic variety. (Decades later, Grauert showed that there are no exotic real analytic manifolds that don’t embed in euclidean space, strengthening “canonically” to “essentially uniquely.”)

Of course, this just pushes the real analytic question from the manifold to the almost complex structure. For some kinds of structures, it really matters, like the Reeb foliation which is C^oo but not real analytic. I don’t know if people want to use bump functions when building their involutive almost complex structures, but the existence of the strong NN theorem suggests not.

2. Kevin Lin - March 7, 2009

Great post! A small correction: the complexified tangent bundle of M is not the tangent bundle of M_C, it is the pullback of the tangent bundle of M_C along the inclusion of M into M_C.

3. John Hubbard - April 11, 2009

May I suggest you look at appendix A4 of the book

Teichmuller theory, Vol 1, by John Hubbard, Matrix editions, 2006. It contains the complete proof sketched above.

With one of my students (Milena Pibiniak), I am working on the C-infinity version, which I believe can be reduced to generalities in “global analysis”, i.e., calculus on Banach manifolds.

If anayone is interested, I can keep them informed about progress on that front.


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