Motivated by this question, 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.
closed as no longer relevant by Deane Yang, Tim Perutz, Johannes Ebert, S. Carnahan♦ Feb 24 '11 at 2:17This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question. 


Given a complex manifold, you have a bundle of (1,0)forms within complexified 1forms which is generated (over $C^\infty$) by differentials of holomorphic functions. This gives a decomposition of 1forms tensor C into (1,0) and (0,1)part. Your I is an operator which is equal to $\sqrt 1$ on (1,0)forms and $\sqrt 1$ on (0,1)forms. It is in fact real, hence defines a real endomorphism of TM, squared to Id. 

