Say we have a complex variety given by:
$$x^5+y^4+3=0$$
with $(x,y)\in \mathbb{C}^2$. Then apparently this is not a Kähler surface. (Or is it?) But if we do the projective completion:
$$x^5+y^4z+3z^5=0$$
With $(x,y,z)\in \mathbb{PC}^2$ then this is a Kähler surface as all complex projective varieties in complex-projective space are Kähler as "Every smooth complex projective variety" is Kähler.
But the second case is just the first case with extra points `at infinity'.
So how can you have a complex structure on the second case but not on the first? The second is just the first projected onto the hyper-plane $z=1$ with some points projected to points at infinity
The first case is non-compact but this does not seem to be a condition of Kähler surfaces. Why is the first case not called Kähler also?