Holomorphic functions of several variables - 1

The theory of holomorphic functions of several variables is the foundation of complex geometry, which uses as its local model. We try to generalize some results from the one variable case, and explore some different phenomena in the several variables case.

Suppose is a point in and satisfies for each . Then we define the polydisc to be the set consisting of the points such that

For a continuous function defined on a connected open subset , we say is holomorphic on if holds on . Meanwhile, we can define for each , called the partial derivative of with respect to .

Applying the result of the one variable case, we obtain the following formula.

Theorem (Cauchy integral formula). Suppoose is a continuous function such that is holomorphic with respect to each single component at each point in . Then for each the following holds

The above formula can be used to obtain the the power series expansion of a holomorphic function. Suppose is a connected open subset and is holomorphic. Then for each , there is a polydisc such that where

The preceding results enable us to generalize the identity theorem, the maximal module principle and the Schwarz lemma to the several variables case.

Theorem (Identity theorem). Suppose is a connected open subset in and is holomorphic. If vanishes on a neighborhood of some , then equals zero on the entire .

Proof: Let be the subset of consisting of the points on a neighborhood of which vanishes. Then is nonempty and open. By the local power series expansion of holomorphic functions, we see that if and only if all derivatives of vanish at , implying that is also closed in . It follows that is exactly the entire .

Theorem (Maximal module principle). Suppose is a connected open subset in and is holomorphic. If attains a maximal value at , then is constant on .

Theorem (Schwarz lemma). Suppose and let . Suppose is holomorphic with , and for each . Then for each , we have where

Proof: For a fixed , let Applying the one-variable Schwarz lemma to the function given by we obtain that

An interesting phenomenon which is not expected for the one variable case is the following theorem.

Theorem (Hartogs’ theorem). Suppose and and satisfy that for each . Then any holomorphic function can be uniquely extended to a holomorphic map .

We refer to the proof of this theorem in Huybrechts’ book.

Lemma. Suppose is an open subset of , is a neighborhood of for some , and is a holomorphic function. Then the function given by is holomorphic.

Proof: Since is compact, we can cover it with finitely many neighborhoods with such that on each neighborhood the power series expansion of converges uniformly, and hence commutes with the integral. This yields a power series expansion of locally.

Now we turn to the proof of Hartogs’ theorem.

Proof of Hartogs’ theorem: For any such that for each , gives a holomorphic function on the annulus .

Consider the Laurent expansion

Then since the lemma implies that is holomorphic for .

Meanwhile, the function is holomorphic for any when , which suggests that when and . It follows from the identity theorem that holds whenever . (This is where we need )

Define This power series converges uniformly on the disc , as the maximal module of each can only be attained on the boundary, and that the power series converges on the annulus.