Calabi-Yau manifolds are usually defined (e.g. Wikipedia) in the analytic setting as those compact complex Kähler manifolds with trivial canonical bundle. (But even Wikipedia says that there are several inequivalent definitions.) I have a couple of questions:
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What is the correct definition (or the most used) in the algebraic category? A Calabi-Yau variety is… ? The only clear condition is that it should be an algebraic complex variety $X$ with $\omega_X\cong\mathcal O_X$. But is it required to be proper, or smooth, or to have $h^i(X,\mathcal O_X)=0$ for $0<i<\dim X$?
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According to that definition, what about (algebraic) examples? I would like to have a bunch of them in mind. There is the famous example of the quintic threefold in $\mathbb P^4$, there are elliptic curves, there is affine space (if, in the definition, we do not require properness). What about Abelian varieties, for instance?
Thank you!
Best Answer
Since this question doesn't have an answer yet, let me try to say something.
I'm not sure how helpful it is to talk about the "correct" definition of Calabi–Yau: different people use different definitions depending on the context, and the important thing is just to be clear about which definition you're using. For example, in birational geometry and minimal model theory, often the main thing one cares about is the numerical properties of the canonical bundle, so it makes sense to allow Calabi–Yau to mean any variety such that $K_X \equiv 0$. By contrast, from the point of view of differential geometry (about which I don't really know anything), one wants to understand the holonomy of a manifold, and then the condition that $h^i(\mathcal{O}_X)=0$ (which for varieties with trivial canonical bundle is, as I understand it, equivalent to the holonomy being exactly $SU(n)$ rather than a proper subgroup) is natural.
Something else that seems relevant to your question is the Beauville–Bogomolov decomposition theorem: this says if $X$ is a compact Kähler manifold with $K_X \equiv 0$, then there is a finite étale cover $\tilde{X} \rightarrow X$ such that $\tilde{X}$ is a product of Calabi–Yau manifolds (in the stricter sense), complex tori, and so-called hyperkähler manifolds. The point is that this shows that strict Calabi–Yaus are still something natural — they are one of the basic building blocks of $K$-trivial Kähler manifolds.
Examples: Of course what counts as an example depends on your definition. If we're allowing the most general definition, then as well as strict Calabi–Yaus and abelian varieties as you mentioned, we also have hyperkähler varieties as mentioned above. The Beauville–Bogomolov theorem then shows that these are (up to finite étale covers and products) all examples.
Examples of strict Calabi–Yaus:
Smooth hypersurfaces of degree $n+1$ in $\mathbf{P}^n$, for $n \geq 3$, including quintic threefolds. Adjunction shows that these have trivial canonical bundle, and the Lefschetz hyperplane theorem shows that the appropriate $h^i(\mathcal{O}_X)$ vanish. More generally, complete intersections of type $(n_1,\ldots,n_k)$ and dimension at least 2 in $\mathbf{P}^n$, where $n_1+\cdots+n_k=n+1$.
More generally, appropriate complete intersections in weighted projective spaces, or in products of projective spaces, give more examples.
I don't know many more. One interesting class of examples I can think of is elliptic Calabi–Yau threefolds, which means those possessing a map to a surface whose general fibre is an elliptic curve. Grassi–Morrison's fibre products of rational elliptic surfaces give some interesting examples here. Similarly, there are Calabi–Yaus threefolds with abelian surface fibrations: here an example is Horrocks–Mumford quintics.
Finally, let me say a little about
Examples of hyperkähler varieties: