Motivated by this [question](https://mathoverflow.net/questions/56182/almost-complex-integrability-and-algebraic-varieties), 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.



Global Definition of the Almost Complex Structure of a Complex Manifold

> 
> the construction of the canonical $J$ for a complex manifold is what I'm interested in
> 
> 
> 


Given a complex manifold, you have a bundle of (1,0)-forms within complexified 1-forms
which is generated (over $C^\infty$) by differentials of holomorphic functions. This gives
a decomposition of 1-forms 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.


