How "frequent" are smooth projective varieties $X$ with trivial canonical bundle $\omega\_X = \bigwedge^d \Omega^1\_{X/k}$?


E.g. for curves $C/k$, the canonical bundle is trivial iff the genus $g(C) = 1$ (elliptic curves). What is the situation like in the higher dimensional case?



How "frequent" are smooth projective varieties with trivial canonical bundle?
If $X$ is a complex projective manifold with trivial canonical bundle, then by a theorem of Bogomolov, there is a finite unramified cover $\tilde X$ of $X$ which decomposes into a product $A\times X\_1\times\ldots\times X\_n\times Y$. Here $A$ is an abelian variety; $X\_i$ are irreducible holomorphic symplectic manifolds (simply-connected, with a unique non-vanishing holomorphic 2-form) and $Y$ is a "strict Calabi-Yau" (simply-connected, with no holomorphic 2-form but a non-vanishing holomorphic top-form). Quite how frequent they are depends on your definition of frequent; for example, it's not known whether there are finitely many or infinitely many deformation types of strict Calabi-Yaus in three dimensions. 


