Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result rather than write down the dull proof myself. Is there any well-written proof in the literature to cite ?
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$\begingroup$ While not really an answer to your request for a reference, a standard (I think) observation is that the proof of the Morse lemma in Milnor's "Morse theory" adapts to the holomorphic case with only slight changes. $\endgroup$– Tim PerutzOct 1, 2013 at 0:13
1 Answer
In my opinion a good reference is the book by Greuel, Lossen and Shustin Introduction to Singularities and Deformations. See in particular Theorem 2.46, page 147. In fact, the authors prove the following more precise result.
Theorem (Morse Lemma). Let $f \in \mathfrak{m}^2 \in \mathbb{C} \{x_1,\ldots ,x_n \}$ be a germ of holomorphic function, having a critical point at the origin $\mathbf{0} \in \mathbb{C}^n$. Then the following are equivalent:
(1) the point $\mathbf{0}$ is a non-degenerate critical point of $f$, i.e. the Hessian matrix $H(f)(\mathbf{0})$ has maximal rank;
(2) $\mu(f)=1$, where $\mu$ denotes the Milnor number;
(3) $\tau(f)=1$, where $\tau$ denotes the Tjurina number;
(4) $f$ is right equivalent to $x_1^2+ \ldots +x_n^2$;
(5) $f$ is contact equivalent to $x_1^2+ \ldots +x_n^2$.
Note that the the statement you are looking for is precisely the equivalence between (1) and (4).