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Let $f: \mathbb{C^n} \rightarrow \mathbb{C}$ be a holomorphic function and such that $0 \in \mathbb{C}$ is a regular value. Then the result to be proven is that the set of zeroes $X := Z(f)$ is in fact a complex manifold of dimension $n-1$.

I do not understand the proof that is given in the book (Complex geometry by David Huybrechts). He used the implicit function theorem to generate an open covering $U_i$ for $X$ and open subsets $V_i \in \mathbb{C^{n-1}}$ with bijective holomorphic maps $g_i: V_i \rightarrow U_i$.

I fail to see how the implicit function theorem is used in this case. I would greatly appreciate if anyone having the book can explain it to me! Thanks.

Soby
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1 Answers1

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Let $p\in X$. Because $f(p)=0$ is a regular value, we can assume wlog that $\frac{\partial f}{\partial z_n}(p)\neq 0$. Applying implicit function theorem we get a holomorphic function $h_p:V_p\subseteq\mathbb{C}^{n-1}\rightarrow W_p\subseteq\mathbb{C}$ such that $f(z_1,\dots,z_{n-1},h_p(z_1,\dots,z_{n-1}))=0$ for all $(z_1,\dots,z_{n-1})\in V_p$ and $p$ lays on its graph.

Take $U_p=(V_p\times W_p)\cap X$. Then $g_p:V_p\rightarrow\mathbb{C}^n$, $$g_p(z_1,\dots,z_{n-1})=(z_1,\dots,z_{n-1},h_p(z_1,\dots,z_{n-1}))$$ is some considered chart, as its image is $U_p$.

You can take the atlas $\{(U_p,g_p^{-1})\}_{p\in X}$.

GlassFlake
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