Abstract Nonsense

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

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

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

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Lemma(Generalized Argument Principle): Let $\Omega\subseteq\mathbb{C}$ be a domain and let $f,g\in\mathcal{O}(\Omega)$. Let $\Gamma$ be a simple closed curve in $\Omega$ and suppose that $f$ has no zeroes on $\Gamma$. Then,

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$\displaystyle \frac{1}{2\pi i}\int_\Gamma g(\zeta)\frac{f'(\zeta)}{f(\zeta)}d\zeta=\sum_{j=1}^{n}g(a_j)$

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where $a_1,\cdots,a_n$ are the roots of $f$ in the interior of $\Gamma$.

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

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$\displaystyle \frac{1}{2\pi i}\int_\Gamma g(\zeta)\frac{f'(\zeta)}{f(\zeta)}=\sum_{\alpha}\text{Res}\left(g\frac{f'}{f},\alpha\right)$

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where the sum runs over the distinct roots of $f$ in the interior of $\Gamma$. We see though that if $\alpha$ is a root of degree $m$ at $f$ then $f(z)=(z-\alpha)^mh(z)$ with $h\in\mathcal{O}(\Omega)$ and $h(\alpha)\ne 0$. We see then that

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$\displaystyle g(z)\frac{f'(z)}{f(z)}=g(z)\frac{h(z)}{h'(z)}+g(z)\frac{m}{z-\alpha}$

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From this it’s evident that

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$\displaystyle \text{Res}\left(g\frac{f'}{f},\alpha\right)=mg(\alpha)$

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from where the conclusion follows. $\blacksquare$

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The operative part of this theorem is the following:

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Corollary: Let $\Omega\subseteq\mathbb{C}$ be open and $f\in\mathcal{O}(\Omega)$. Suppose that $\Gamma$ is a simple closed curve in $\Omega$ such that $f$ has one zero $\alpha$, which is simple, in the interior of $\Gamma$ and no roots on $\Gamma$.Then,

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$\displaystyle \alpha=\frac{1}{2\pi i}\int_\Gamma \zeta\frac{f'(\zeta)}{f(\zeta)}\;d\zeta$

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Now, we can pretty easily prove the holomorphic IFT:

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Proof(Holomorphic IFT): Now, since $F_v$ is continuous (remember, Hartog’s theorem!) there exists an $\varepsilon>0$ such that $|w|,|v|<\varepsilon_1$ implies that $F_v(w,v)\ne 0$. Moreover, since $F_v(0,0)\ne 0$ there exists a holomorphic function $g(y)$ not vanishing at the origin such that $F(0,v)=vg(v)$. Thus, we see that there is some $\varepsilon_2>0$ such that $F(0,v)$ has only the simple zero $(0,0)$ as a root for $|yv<\varepsilon_2$. Let $\varepsilon=\text{min}\{\varepsilon_1,\varepsilon_2\}$ and define

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$\displaystyle \eta(x)=\frac{1}{2\pi i}\int_{|v|=\varepsilon}\frac{F_v(w,v)}{F(w,v)}\; dv$

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Now, the Argument Principle tells us that $\eta$ is a continuous mapping $D_\varepsilon(0)\to\mathbb{Z}$. By construction $\eta(0)=1$, and thus we see that $\eta$ is identically $1$ on $D_\varepsilon(0)$. Thus, for each $w\in D_\varepsilon(0)$ there exists a unique solution to $F(w,v)=0$, call this $v$ value $\varphi(w)$. By the corollary above, we see that

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$\displaystyle \varphi(w)=\int_{|y|=\varepsilon}v\frac{F_v(w,v)}{F(w,v)}\;dv$

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and is thus holomorphic. In particular, we see that $\varphi$ is a holomorphic mapping $D_\varepsilon(0)\to\mathbb{C}$ such that on the polydisc $D_\varepsilon(0)\times D_\varepsilon(0)$ one has that $(w,\varphi(w))$ is the unique solution to $F(w,v)=0$. $\blacksquare$

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Remark: Note that choosing $v$ as the non-zero partial played no role and the above is true if we require that $F_w(0,0)\ne0$.

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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 $f:U\to\mathbb{C}$ where $U\subseteq\mathbb{C}^2$ to be a point $y\in\mathbb{C}$ such that for every $p\in f^{-1}(y)$ we have that $(f_w(p),f_v(p))\ne(0,0)$. We call a function $f\in\mathcal{O}(U)$ non-singular if $0$ is a regular value for $f$. We then have the following theorem:

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Theorem: Let $U\subseteq\mathbb{C}^2$ be open and $f\in\mathcal{O}(U)$ be non-singular. Then, $Z(f)$ is a complex manifold of dimension one.

Proof: Let $p=(w_o,v_0)\in Z(f)$ be arbitrary. By assumption, one of $f_w(p),f_v(p)\ne 0$. Thus, the holomorphic IFT tells us that there exists a neighborhood $W_p$ of $w_0$ in $\mathbb{C}$ and a holomorphic function $g_p:W_p\to\mathbb{C}$ such that $\pi_{i_p}^{-1}(W_p)$ is just the graph of $g_p$ over $W_p$, where $i_p\in\{1,2\}$ depends on which partial is nonzero. We claim that $\left\{(\pi_{i_p}^{-1}(W_p),\pi_1)\right\}$ is an atlas for $Z(f)$. 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 $p_1,p_2\in Z(f)$ such that $\pi_{i_{p_1}}^{-1}(W_{p_1})\cap\pi_{i_{p_2}}^{-1}(W_{p_2})$ contains a point $p$. Now, if $i_{p_1}=i_{p_2}$, say they are both equal to $1$, then obviously the transition map just projects down and lifts back up–in other words, is the identity. Now, if $i_{p_1}\ne i_{p_2}$, say $i_{p_j}=j$ then one can verify fairly easily that the overlap map is just $g_p(z)$, which is holomorphic. The conclusion follows. $\blacksquare$

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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.