On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is closed, but gives a nonzero integral for your form, then you know the form is not $d$-exact.
However, I can't seem to find an equivalent test for $\overline\partial$-exact forms. My intuitive understanding of the complex vector space is a bit shaky, but it I think if we integrate $\overline\partial f$ in a closed loop, we will get $-\int\partial f$ in that loop, which we don't really know anything about. We can't come up with a loop on which $\int\partial f$ is necessarily zero, or necessarily anything specific.
Is there some way to salvage this method, or is there another way we can prove $\overline\partial$-exactness? Any overview about how you'd approach this sort of problem is also welcome.