nLab
Calabi-Yau variety

Idea

An $n$-dimensional Calabi-Yau variety is an $n$-dimensional Kähler manifold with (holomorphically, rather than topologically) trivial canonical bundle. This is equivalent to say that it is an even dimensional real Riemannian manifold of dimension $2N$ whose holonomy is in the subgroup $\mathrm{SU}(N)\subset O(2N,ℝ)$.

For compact Kähler manifolds, Yau's theorem? (also known as the Calabi conjecture?) implies that the above conditions are also equivalent to the vanishing of the first Chern class.

Is it also true for non-compact?

Note that $c_1(X)=0$ implies in general that the canonical bundle is topologically trivial. But if $X$ is a compact Kähler manifold, $c_1(X)=0$ implies further that the canonical bundle is holomorphically trivial.

The language used in this article is implicitly analytic, rather than algebraic. Is this OK? Or should I make this explicit?

Generalized Definition

A Calabi-Yau variety can be described algebraically as a smooth proper variety $X$ of dimension $n$ over a field $k$ (not necessarily algebraically closed and not necessarily of characteristic $0$) in which ${\omega }_X={\wedge }^n{\Omega }^1\simeq {𝒪}_X$ and also $H^j(X,{𝒪}_X)=0$ for all $1\le j\le n-1$.

If the base field is $ℂ$, then one can form the analyticification of $X$ and obtain a compact manifold that satisfies the first given definition.

Examples

• In dimension 2: K3 surface.

Related concepts

classification of special holonomy manifolds by Berger's theorem:

G-structure	special holonomy	dimension	preserved differential form
Kähler manifold	U(k)	$2k$	Kähler forms
Calabi-Yau manifold	SU(k)	$2k$
hyper-Kähler manifold	Sp(k)	$4k$
G2 manifold	G2	$7$	associative 3-form
Spin(7) manifold	Spin(7)	8	Cayley form

• Calabi-Yau object

  • Calabi-Yau algebra, Calabi-Yau manifold, generalized Calabi-Yau manifold

• supersymmetry and Calabi-Yau manifolds