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 ;)
