# Is there a holomorphic Morse-Bott lemma?

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 $s = {\rm dim} S$?

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?

• But this is simply stated in a thesis without proof, so it would behoove us to not accept this. Sep 21 '14 at 1:49
• It should just be a restatement in different language. It is rigorously proved in multiple different texts but in more complicated language. Additionally, I would be very surprised if they let a PhD student simply state something like that in a successfully submitted thesis without proof or reference unless it was generally accepted to be true. My supervisor/external would have words to say if I did that. Sep 21 '14 at 11:29

The appropriate notion in the world of complex manifolds is the notion of a Lefschetz fibration on a Lefschetz pencil. To my knowledge, there is a nice exposition of the theory in

Liviu Nicolaescu, An invitation to Morse theory.

See Chapter 4 in the first edition of that book or Chapter 5 in the second edition. I think Theorem 4.3 or Theorem 5.3, resp., should be the result that you are searching for.

• Thanks. But I don't find anything related to my question.
– UVIR
Apr 20 '14 at 19:06