Morse lemma for holomorphic functions

Question

If $f:C^n\to C$ is holomorphic in a neighbourhood of $0$ and $0$ is a nondegenerate critical point, then there is a neighbourhood $U$ of $0$ with a holomorphic local chart, namely a holomorphic invertible map $\varphi =(w_1,…,w_n):U→V\subset C$, such that $\varphi(0)=0$ and $=f∘φ^{−1}$ takes the form $f(w)=f(0)+w^2_1+…+w^2_n$. Is there a good reference (understandable with the elementary knowledge of differential topology) to this? It would be really helpful if somebody could write the proof?

Answer 1

The following proof comes from the book "The monodromy group" by Henryk Żołądek (the author says that the proof was suggested by T. Maszczyk).

First you need the lemma of Hadamard (see wikipedia for the proof which can be easily adapted to the complex case) :

Hadamard's lemma : Let $f : \Bbb C^n \to \Bbb C$ be holomorphic with $f(0) = 0$. Then, there exists holomorphics maps $g_1, \dots, g_n : \Bbb C^n \to \Bbb C$ so that $f = \sum x_ig_i(x)$.

Now we can prove the complex Morse lemma.

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Proof of the Morse lemma :

Let $f : \Bbb C^n \to \Bbb C$ be holomorphic with $f(0) = 0$, $d_0f = 0$ and a non-degenerate hessian at $0$. We can write $f(x) = \sum x_ig_i(x)$ by Hadamard lemma. By hypothesis $g_i(0) = 0$ so we can apply Hadamard lemma again, and we can write $$ f(x) = \sum_{i,j} x_ix_j g_{ij}(x) $$

Writing $h_{ij} = g_{ij} + g_{ji}$, we obtain that the matrix $H(x)$ with coefficients $h_{ij}(x)$ is symetric. Moreover, $H$ is not singular for small $x$, because $H(0)$ is the Hessian of $f$ at $0$.

Now we look at the quadratic form $q_x(\xi) = \sum h_{ij}(x)\xi_i \xi_j $. The form $q_x$ is diagonalisable at the origin, so by implicit function theorem there are local holomorphic coordinates $\eta_i = \sum_j \alpha_{ij}(x)\xi_i$ so that $q(\eta) = \sum_i \eta_i^2$. Now we can take $z_i = \sum_j \alpha_{ij}(x)x_j$.