Motivated by this [question][1], I began to wonder if there is a global definition of the almost complex structure of a complex manifold. It is (almost) always presented as multiplication by complex $i$ on the tangent space, and then globalized. Using the  formulae given earlier
$$
\overline{\partial}\omega = \frac{1}{2}(\text{d}\omega + i \text{d}(J\omega)), 
$$
and
$$
\partial \omega = \frac{1}{2}(\text{d}\omega - i \text{d}(J\omega)), 
$$
it is easy to see that 
$$
-\frac{i}{2}d\omega = d(J\omega).
$$
Thus, $J\omega = -\frac{i}{2}\omega + \omega'$, where $\omega'$ is some closed form. What this $\omega'$ is, however, I cannot see.


  [1]: http://mathoverflow.net/questions/56182/almost-complex-integrability-and-algebraic-varieties