I am looking for a characterization of (0,1)-forms on a CR manifold M that admit a real primitive, i.e. those can be written as:
for a real function f.
If M is a complex manifold, by expanding ddf=0 one obtains the following characterization:
$\overline\partial\omega=0$, $\partial\overline\omega=0$, $\partial\omega+\overline\partial\overline\omega=0$.
In the CR case however, there is not a good substitute for $\partial$, and also the symmetry between (0,1) forms and (0,1)-forms fails.
Edit: One easy condition is $\overline\partial_M\omega=0$. In general it is a difficult problem even to know say if $\omega=\overline\partial_M g$ for some complex function g.
My question should be rephrased as follows: Assuming that there exist a complex solution g to $\omega=\overline\partial_M g$, when is it possible to choose g real?