# Is there any topological information encoded by the zero locus of a complex Hessian?

On a Kahler manifold one can always find a function $K$ so that (up to some constant) the Kahler form $\omega$ can be written as $\omega = \partial \bar{\partial}K$ (sometimes, under conditions that I don't quite know the full justification for but are probably a key part of any suitable answer to this question, this statement only holds true locally). By definition, the Kahler form is nondegenerate implying that $\partial \bar{\partial}K \neq 0$ everywhere, except perhaps at certain points where the above statement fails globally. I am therefore wondering whether anyone has come across a situation where using a converse of this result one can define a topological invariant of complex manifolds in the following sense:

1) On a complex manifold $M$, choose a function $F$ satisfying suitable conditions so that $Z$ (defined below) is finite, compact etc. in analogy with usual Morse theory.

2) Construct the form $\Omega = \partial \bar{\partial} F$ and consider its zero locus $Z \subset M$.

3) Construct the analogue of a Morse-Witten complex using elements of $Z$ and signed counts of some appropriate objects "connecting critical points of $\Omega$", whatever they might be!

Intuitively, the homology of this complex might provide some topological information about $M$, in particular obstructions to the existence of a Kahler form or when we can have $\omega = \partial \bar{\partial}K$ globally.