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 is an endomorphism of the tangent bundle such that .
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 vector field. Actually, David discussed forms, but vector fields are just the dual notion. Namely they are those complex valued vector fields which are eigenvectors for with eigenvalue .
The N-N theorem can be stated as follows:
An almost complex structure is integrable iff the vector fields on 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 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 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 . This is a complex manifold, whose complex dimension is the same as the real dimension of . It is given by an atlas whose transition functions are the same power series as those for , except they are now viewed complex analytic functions.
Now the complexified tangent bundle of is the actual tangent bundle of and so the subspace of vector fields gives a holomorphic distribution on (here we also assume that the original complex structure was real analytic). Then we apply a holomorphic version of the Frobenius theorem to show that locally we can find a holomophic map whose kernel is this distribution. Then we show that the restriction of this holomorphic map to is a diffeomorphism. By definition, this map will be -holomorphic and thus we build an atlas on . This completes the proof. Pretty nifty, eh?