## Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. II)

**Point of Post: **This is a continuation of this post.

The proof the of the holomorphic IFT follows from the following lemma, which is a slight generalization of the Argument Principle:

**Lemma(Generalized Argument Principle): ***Let be a domain and let . Let be a simple closed curve in and suppose that has no zeroes on . Then,*

*where are the roots of in the interior of .*

**Proof: **This is done by merely applying the residue Residue theorem. Namely, we know that:

where the sum runs over the distinct roots of in the interior of . We see though that if is a root of degree at then with and . We see then that

From this it’s evident that

from where the conclusion follows.

The operative part of this theorem is the following:

**Corollary: ***Let be open and . Suppose that is a simple closed curve in such that has one zero , which is simple, in the interior of and no roots on .Then,*

Now, we can pretty easily prove the holomorphic IFT:

**Proof(Holomorphic IFT): **Now, since is continuous (remember, Hartog’s theorem!) there exists an such that implies that . Moreover, since there exists a holomorphic function not vanishing at the origin such that . Thus, we see that there is some such that has only the simple zero as a root for . Let and define

Now, the Argument Principle tells us that is a continuous mapping . By construction , and thus we see that is identically on . Thus, for each there exists a unique solution to , call this value . By the corollary above, we see that

and is thus holomorphic. In particular, we see that is a holomorphic mapping such that on the polydisc one has that is the unique solution to .

*Remark: *Note that choosing as the non-zero partial played no role and the above is true if we require that .

Now, from this we can prove that loci of certain holomorphic functions are actually Riemann surface. In particular, like in the case of smooth manifolds we define a *regular value *of a holomorphic map where to be a point such that for every we have that . We call a function *non-singular *if is a regular value for . We then have the following theorem:

**Theorem: ***Let be open and be non-singular. Then, is a complex manifold of dimension one.*

**Proof: **Let be arbitrary. By assumption, one of . Thus, the holomorphic IFT tells us that there exists a neighborhood of in and a holomorphic function such that is just the graph of over , where depends on which partial is nonzero. We claim that is an atlas for . Indeed, it’s clear that this is a topological atlas, and so it suffices to check that the elements of the atlas are compatible. To this end, suppose that we have two points such that contains a point . Now, if , say they are both equal to , then obviously the transition map just projects down and lifts back up–in other words, is the identity. Now, if , say then one can verify fairly easily that the overlap map is just , which is holomorphic. The conclusion follows.

**References: **

[1] Varolin, Dror. *Riemann Surfaces by Way of Complex Analytic Geometry*. Providence, RI: American Mathematical Society, 2011. Print.

[2] Miranda, Rick. *Algebraic Curves and Riemann Surfaces*. Providence, RI: American Mathematical Society, 1995. Print.

[3] Forster, Otto. *Lectures on Riemann Surfaces*. New York: Springer-Verlag, 1981. Print.

[4] Conway, John B. *Functions of One Complex Variable*. New York: Springer-Verlag, 1978. Print.

[5] Gong, Sheng. *Concise Complex Analysis*. Singapore: World Scientific, 2001. Print

[6] Hörmander, Lars. *An Introduction to Complex Analysis in Several Variables*. Princeton, NJ: Van Nostrand, 1966. Print.

[7] Krantz, Steven G. *Function Theory of Several Complex Variables*. New York: Wiley, 1982. Print.

[8] Ohsawa, T. *Analysis of Several Complex Variables*. Providence, RI: American Mathematical Society, 2002. Print.

[9] Ebeling, Wolfgang. *Functions of Several Complex Variables and Their Singularities*. Providence, RI: American Mathematical Society, 2007. Print.

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