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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?

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up vote 14 down vote accepted

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.

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It's worth noting that to make some strict Calabi-Yau's at home, it suffices to take a smooth degree $n+2$ hypersurface in $\mathbb P^{n+1}$. This fabricates cheap examples of non-trivial families of such manifolds in any dimension. – Gunnar Þór Magnússon Nov 25 '11 at 4:30

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