Reference request for Morse theory with complex valued functions

Question

I was just wondering if there is a version of Morse theory by considering maps from $f: M \to \mathbb{C}$ where $M$ is a complex manifold and $f^{\prime \prime }(z) \neq 0 $ whenever $f^{\prime}(z)=0$.

After searching for some time, I found that functions to a circle have been studied but I could not find anything for a complex valued function. Does anyone know of any references or does this simplify to Morse theory in the real case? Thank you.

Answer 1

The correct holomorphic analogue of Morse functions on compact complex manifolds is a Lefschetz pencil (see also and references therein); see also here. A LP is a holomorphic map $f: M\to {\mathbb C} P^1$, which has only Morse critical points. They are well-studied.

Edit (thanks to Nicolas Hemelsoet): 1. See also "The topology of complex projective varieties after S. Lefschetz", by Klaus Lamotke.

2. One can also consider holomorphic maps to Riemann surfaces of higher genus ($\ge 1$), but they do not always exist (but when they do exist, they become quite useful). Maybe one should regard them as holomorphic analogues of circle-valued Morse functions.