One way of stating Newlander-Niremberg's theorem is by saying an almost complex manifold $(M,J)$ is complex $\Leftrightarrow \overline{\partial}^2=0$.
I'm confused by this, because I can't see why wouldn't $\overline{\partial}^2=0$ be always true.
Let a neighbourhood $U\subset M$ with coordinates $x_1,y_1,...,x_n,y_n$. Then a $k$-form $\omega\in\Omega^{\ell,m}$ will be like: $$\omega=\sum_{|I|+|J|=k}f_{I,J}dz_I\wedge d\overline{z}_J$$
where $dz_j=dx_j+idy_j$ and $d\overline{z}_j=dx_j-idy_j$. Then we would have:
$$\overline{\partial}\omega=\sum_{j}\frac{\partial f_{I,J}}{\partial \overline{z}_j}d\overline{z}_j\wedge dz_I\wedge d\overline{z}_J$$ $$\overline{\partial}^2\omega=\sum_{k}\sum_{j}\frac{\partial^2 f_{I,J}}{\partial\overline{z}_k\partial \overline{z}_j}d\overline{z}_k\wedge d\overline{z}_j\wedge dz_I\wedge d\overline{z}_J$$
As the $d\overline{z}_j$ and $d\overline{z}_k$ are permuted, we get opposite signs, so everything's canceled out and $\overline{\partial}^2=0$, regardless of $\omega$.
Obviously I'm doing something wrong. Somewhere, I must have assumed implicitly that $M$ is complex, but I can't see where that happened.