Let $J$ be an almost complex structure on an algebraic variety $V$. As we all know, $J$ comes from a complex structure if the Nijenhuis tensor of $J$ vanishes. What I would like to know is if there exists a simpler characterisation of integrability than this for varieties (as opposed to general manifolds). 



Almost Complex Integrability and Algebraic Varieties
If you want equivalent conditions to the Nijenhuis tensor vanishing then one is that the induced $\bar \partial$ operator defines a complex, i.e. that $\bar \partial^2 = 0$. Another one is that the exterior derivative decomposes as $d = \partial + \bar \partial$. If you want explicit examples of almost complex manifolds that are not complex, that's going to be more difficult, see the answers to tinyurl.com/4zrkar6


To find the $\bar \partial$ operator associated to an almost complex structure $J$ on a smooth manifold $M$, one needs to note that $J$ induces a splitting $T\_M \otimes \mathbb C = T^{1,0} \oplus T^{0,1}$ of the tangent bundle into $i$ and $-i$ eigenvectors (the eigenspaces being marked with $(1,0)$ and $(0,1)$, respectively).


The same thing happens on the level of 1-forms (and indeed on the level of $k$-forms): they split into $(p,q)$-forms like on complex manifolds. If $\pi^{p,q} : \bigwedge^k T\_M \to \bigwedge^{p,q} T\_M$ is the projection onto the space of $(p,q)$-forms, then the $\bar \partial : \bigwedge^{p,q} T\_M \to \bigwedge^{p,q+1} T\_M$ operator associated to $J$ is $\bar \partial\_J = \pi^{p,q+1} \circ d$.


Once one does the calculations this comes out to
$$ \bar \partial \alpha = \frac 1 2 \left( d \alpha + i d J \alpha \right) $$
for a $(p,q)$-form $\alpha$. A similar formula holds for the $\partial$ operator, you just have to change $i$ to $-i$. (I may have confounded the signs here.)


A very good reference for the linear algebra parts (i.e. most of this) is Chapter 2 of Huybrecht's "Complex geometry".


For the conditions equivalent to the vanishing of the Nijenhuis tensor I seem to remember the first chapter of <http://tinyurl.com/4cqspu7> Moroianu's notes on Kahler geometry being very helpful when I went through this a couple of months ago.


Finally, on why the vanishing of the Nijenhuis tensor implies that we indeed have a complex structure I recommend Demailly's book - Chapter 8, section 11 (page 396) has all the details:
<http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf>


