Dimension-specific phenomena in algebraic geometry In differential topology, there are some funny phenomena that can only happen in dimension 4. For example, only in dimension 4 you can have a closed topological manifold admitting infinitely many distinct smooth structures. 
Is there something like this happening in algebraic geometry? Does there exist an integer $k>1$ and some interesting geometric statement that only holds for varieties of dimension$\neq k$? 
P.S. It should be noted that a professional topologist has said that he has doubts about one important paper on manifold topology. He did not give a lot of details, and not being a topologist, I can not judge the veracity of his claims. I do not know if the existence statement in the second sentence of this post can be established independently of that paper. If somebody does know, let us know. 
 A: Let $X,L$ be a smooth polarized projective variety of dimension $n$ with $K_X =\mathcal{O}_X$. Let $v \in H^{\bullet}(X,\mathbb{Q})$ a primitive vector and consider $M_{L,v}$ the moduli space of $L$-stable sheaves with Chern character equal to $v$. Assume that the actual dimension of $M_{L,v}$ is equal to its virtual dimension. Then $M_{L,v}$ has an open subset which is smooth, say $M_{L,v}^{sm}$.
Miracle when $\dim X =2$ : $M_{L,v}^{sm}$ carries a holomorphic symplectic form! This is because only when $\dim X = 2$ is there a symplectic isomorphism $\mathrm{Ext}_X^1(F,F) \simeq \mathrm{Ext}_X^{1}(F,F)^*$ (induced by Serre duality).
For the time being, there is no Theorem which implies that all moduli spaces of sheaves (or rather objects) on higer dimensional projective varieties that carry a holomorphic symplectic form are necessarily moduli spaces of objects on a $\mathrm{K}$-trivial surface. 
In fact, we know examples of such moduli spaces which original constructions go via higher dimensional manifolds (the Fano scheme of lines on a cubic fourfold for instance). But in every such example known, there is a (possibly non commutative) $\mathrm{K}$-trivial surface hidden in the story. For instance, the Fano scheme of lines on a cubic fourfold can be realized as the moduli space of objects in the derived category a non-commutative $\mathrm{K}$-trivial surface which sits inside the derived category of the cubic fourfold.
So it seems that having moduli spaces carying holomorphic symplectic form is a miracle related to (possibly non-commutative) two-dimensional $\mathrm{K}$-trivial varieties of dimension 2.
Note the miraculous numerical coincidence of my answer with your question : complex dimension $2$ corresponds to real dimension 4 ;)
