# Definition and examples of Calabi-Yau varieties

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:

1. 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 $0<i<\dim X$?

2. 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!

• Abelian varieties do have $\omega_X\simeq\mathcal{O}_X$, so if this is the only requirement then yes, they are Calabi-Yau varieties (moreover they are compact). Another example would be $K3$ surfaces. Nov 14, 2013 at 16:24
• All of those conditions are typically required: smooth, proper, $\omega_X\simeq \mathcal{O}_X$, and $h^i(\mathcal{O}_X)=0$ for $0<i<\dim X$. Thus, abelian varieties are usually not considered to be Calabi-Yau in dimension $\geq 2$.
– Matt
Nov 14, 2013 at 17:25
• Oh, so in the "typical" definition there are more conditions than I expected. Are there any others? Also, what lies behind the requirement that $h^i(\mathcal O_X)$ vanishes? Nov 14, 2013 at 22:28
• I'm not sure about the history, but I assume that vanishing condition came out of years of trying different definitions and this one sort of stuck as being particularly useful. One condition that appears with some frequency, but not a ton is to require $X$ to also be simply connected. This is because for $X$ a K3 surface, the previous conditions imply this.
– Matt
Nov 15, 2013 at 20:14
• @AsalBeagDubh: very nice answer and instructive for me:) Nov 18, 2013 at 22:14

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:

1. 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 $n_1+\cdots+n_k=n+1$.

2. More generally, appropriate complete intersections in weighted projective spaces, or in products of projective spaces, give more examples.

3. 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:

1. Very few are known. See this MO question.
• This is a great answer! In Horrocks-Mumford I cannot find the condition that the general fiber is an elliptic curve, why is this true? does it somehow follow from the construction? Nov 18, 2013 at 20:48
• Glad you liked it! Sorry, HM quintics are fibred by abelian surfaces. I'll fix that in a moment...
– user64687
Nov 18, 2013 at 21:15
• Dear @AsalBeagDubh, do you know the reference where the Beauville–Bogomolov decomposition theorem was first proved? On Wikipedia (en.wikipedia.org/wiki/Fedor_Bogomolov) there are several papers with more or less the same promising title (and only Bogomolov's name, though), but I cannot access them unfortunately. Also, do you know a textbook where I can find a proof of the result? May 7, 2014 at 21:25
• Dear @Brenin, these slides of a talk by Verbitsky bogomolov-lab.ru/DC-2011/talks/verbitsky.pdf indicate that the theorem was stated in the paper The decomposition of Kähler manifolds with a trivial canonical class. *(Russian) Mat. Sb. (N.S.) 93(135) (1974), 573–575) Beauville's article Variétés Kähleriennes dont la première classe de Chern est nulle. (J. Differential Geom. Volume 18, Number 4 (1983), 755-782) appears to give a complete proof.
– user64687
May 8, 2014 at 7:59