Complex manifolds - 2

In this post we first study some properties of a holomorphic map from its Jacobian, and then introduce the concept of submanifolds and subvarieties. After that we disuss a bit about differential forms, which is a preparation for calculus on a complex manifold.

Inverse function thorem and implicit function theorem

In analogy to the real case, we have the following two standard results.

Theorem (Inverse function theorem). Suppose is an open subset and is holomorphic. If is nonsingular at , then there exists a neighborhood containing and a neighborhood containing such that is a holomorphic homeomorphism.

Proof: Since at , the real inverse function theorem yields a smooth inverse of near . It remains to show the holomorphic property of . As , we have for any . It follows from the nonsingularity of that for each and , implying that is holomorphic.

Theorem (Implicit function theorem). Suppose is an open subset, and is holomorphic. If satisfies that then there exist open subset and a holomorphic map such that and

Proof: Again the real implicit function theorem yields a smooth function satisfying the required property. To show the holomorphic property, note that for , which implies for any and .

However, we also have some special features of the complex case.

Theorem. Suppose is a bijective holomorphic map between two open subsets . Then is nonvanishing. In particular, is a holomorphic homeomorphism.

Proof: Prove by induction on . The case when is proved in the reviewing post of holomorphic functions of one variable. Suppose the assertion is proved for any . Consider any such that . We claim that . Assume that . Then we may suppose that is nonsingular. By the inverse function theorem, form a local coordinate system around . It is clear that maps bijectively to . However, the Jacobian of the restriction of to is singular at , which contradicts the induction hypothesis. We conclude that we must have .

By the above discussion, we see that is constant on each connected component of . Since Weierstrass preparation theorem tells us that has positive dimension locally if it is nonempty, the injectivity of implies that is nonvanishing on .

Submanifolds and subvarieties

Like the real case, we can consider submanifolds of a compplex manifold. Suppose is a complex manifold of dimenional . Then a -dimensional complex submanifold is a subset of satisfying that there is a collection of holomorphic coordinate charts of covering such that for each ,

By the inverse function theorem, we can see that this is equivalent to that is given by the zero sets of holomorphic functions such that .

The idea of express a subset as the zero set of some holomorphic functions gives us the concept of subvarieties. An analytic subvariety of a complex manifold is a subset given locally as the zero set of a finite collection of holomorphic functions. A point is called a smooth point or a regular point if is given in a neighborhood of by holomorphic functions with . Denote the set of regular points on by , and let . The points in are called sigular points of . An analytic variety is irreducible if it cannot be written as the union of two proper analytic subvarieties.

We can see that each connected component of is a complex submanifold . There is a theorem saying that an analytic variety is irreducible if and only if is connected. Thus we can define the dimension of an irreducible variety to be the dimensional of .

Differential forms on a complex manifold

The last part of this post devotes to some discussion of -forms on a complex manifold . Viewing as a smooth manifold, we can consider the space of -forms on . Let Then the exterior differentiation gives a linear map

For each point , the decomposition

induces a decomposition

Correspondigly, we obtain the decomposition where consists of -forms satisfying for each . A form is said to be of type , and is also called a -form on .

Let be the projection of onto . Define Using local coordinates, we may consider a -form and direct computation yields that and

We can verify that the operators and have the following properties:

• ;
• and ;
• for and , we have

We see that and have the similar properties to the exterior differentiation . In next post, they will be used to build a holomorphic analogy to the de Rham cohomology theory.