# (Alm) Complex Structures and things being holomorphic

I’m learning some complex geometry and this post is about some basics I re-organized for myself. Most of the material is coming from chapter 1 Huybrechts book ‘Complex Geometry.’

Let be a complex manifold. It is in particular a real manifold of even dimension. Not every even dimensional real manifold is a complex manifold; I believe is an example. This lead to the question:

When can an even dimensional real manifold be made into a complex manifold?

The short answer to this question is: whenever has a *complex structure*. A complex structure is an *almost complex structure *such that the local charts are *holomorphic*.

Again is a real manifold and is a real vector bundle on . A complex structure on the vector bundle is a vector bundle endomorphism such that . Note any real bundle admitting a complex structure must have even fiber dimension basically because it can be made into a vector space.

An almost complex structure on is a complex structure on the tangent bundle .

1. Using this idea, for any even dimensional manifold the local charts can be made into charts.

Locally is say with basis . Relabel and . Define a map via and . Note and . Together has the structure of a -vector space via . So general theory says . The isomorphism is exactly

2. For there is a natural complex structure.

Note is trivial, i.e. . So define via . In real coordinates this is

this is just the above example in slightly different notation.

3. A very important aspect of bundles with complex structures is that they can be tensored with . The significance is that each fiber is a vector space with two complex structures:

This is very important because it gives rise to the decomposition

(0)

note and . In effect the complex structure naturally picks out a subspace that behaves linearly and another that behaves anti-linearly, analogous to .

This decomposition allows me to make sense of the notion of forms and the operators .

Thinking of the tangent bundle, i.e , the decomposition passes to the dual vector space . Consequently I can declare forms to be

Its a linear algebra fact that

(1)

Recall there is the exterior derivative on sections of the exterior powers of the cotangent bundle (I’m using to denote just smooth sections)

In local real coordinates of a section is and

Define and define similarly. -linear extension gives a map which locally is

Since I can project to any of the subspaces in (1) I can define by the composition

and similarly for . I’m use to it being true that but this only happens for nice almost complex structures. A not so nice one is presented below.

Notice applying the above decomposition for the case of gives

(2)

The 1/2 is in there so that it works out where

**Holomorphicity**

Let be a smooth map where and both open. Let be the complex structures on . Let be the differential of the map. Then is holomorphic if .

This can be motivated from the 1-dim case. In this case holomorphic means . In real coordinates this is the same as Cauchy Riemann equations:

The logic is iff the Cauchy-Riemann equations iff iff is holomorphic in sense of the definition above.

In more cumbersome language a function is a map of (real) manifolds:

The maps on (real) tangent spaces is given by the Jacobian matrix i.e. . For the sake of notation write this as

Assuming the Cauchy-Riemann equations (for the third line):

where I note that . This argument is reversible. Thus is equivalent to and similarly for , i.e. preserves the decomposition (0). The last implication in covered in the general case.

Now going to the general case I see if then

Conversely, if preserves (0) then

but noting that given by is an isomoprhism implies in the above that and as desired. So is a good way to generalize holomorphicity.

An almost complex structure that doesn’t give a complex structure.

Look at and use for the tangent space. Define

It can be checked if we map to itself via the identity and given one of them this funky almost complex structure then calling one and the other , then .

In normal complex coordinates i’m used to , but in the above formalism so i.e

but with the funky almost complex structure so . So in perhaps poor notation this is something like

All sorts of bad things happen with this almost complex structure. For example if you look at and apply then you’ll get a component in , i.e. .

In fact is another characterization of the but this is I think difficult to show and is basically covered by a theorem of Newlander and Nierenberg.

In the situation of a manifold with almost complex structure on a vector bundle , I can also consider the extra structure of having an inner product on each fiber.

Some terminology. is compatible with if . The fundamental form associated to is . Together these determine a hermetian form . Its clear any two of determine the third.

Turns out is a real valued 2 form, i.e. and lies in . Notice the importance between differentiating whether or not I tensored with . This is probably a good thing to check in local coordinates. I wont do it now.

This post is already too long, but I’ll mention in passing defines an operator called the Lefshcetz operator on the exterior algebra via wedging with . There is a dual operator of with respect to called , i.e. . With these guys give a representation of .

## About this entry

You’re currently reading “(Alm) Complex Structures and things being holomorphic,” an entry on Math Meandering

- Published:
- March 30, 2010 / 11:58 pm

- Category:
- alg. geo.

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