I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(O_X)$, $h^0(\Omega_X)$, and first betti number.

The only thing I know is that for surfaces, we have $h^1(O_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality hold true for general compact complex manifold?

Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

Thanks to an anonymous friend's comment. On page 444 of Griffiths and Harris' book , they gave an example (Iwasawa manifold) of a non-closed holomorphic 1-form.