TITLE: Is there a holomorphic Morse-Bott lemma?
QUESTION [11 upvotes]: It asks for a generalization of the question in the post
Normal form for a holomorphic Morse function
Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical points, i.e., $df$ vanishes along a complex submanifold $S\subset M$ with nondegenerate Hessian in the normal direction to $S$. For each $p \in S$, can we find a holomorphic local coordinates $z_1, \ldots, z_s, z_{s+1}, \ldots, z_m$ such that locally
$$f(z_1, \ldots, z_m) = \sum_{j = s+1}^m z_j^2 + C,$$
where $s = {\rm dim} S$?

REPLY [2 votes]: An invitation to Morse theory is very good, but it can be a bit heavy. I managed to find a very simple version of the lemma that is directly in the form you requested (I recently made use of this myself in my thesis).
It is a link to the PhD thesis of Matthew Petro, University of Wisconsin, from 2008. A preview is on available on google books, and the Holomorphic Morse-Bott Lemma is Lemma 3.8 in the text; the link should take you straight to it.
Out of curiosity, what are you using it for?