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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:

$\omega=\overline\partial_M f$

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?
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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:

$\omega=\overline\partial_M f$

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 to know 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?
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Real primitive of a complex form on a CR manifold

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:

$\omega=\overline\partial_M f$

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.