nLab
Calabi-Yau variety

Contents

Idea

An nn-dimensional Calabi-Yau variety is an nn-dimensional Kähler manifold with (holomorphically, rather than topologically) trivial canonical bundle. This is equivalent to saying that it is real Riemannian manifold of even dimension 2N2 N which has special holonomy in the subgroup SU(N)O(2N,)SU(N)\subset O(2 N, \mathbb{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)=0c_1(X) = 0 implies in general that the canonical bundle is topologically trivial. But if XX is a compact Kähler manifold, c 1(X)=0c_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?

Definition

A Calabi-Yau variety can be described algebraically as a smooth proper variety XX of dimension nn over a field kk (not necessarily algebraically closed and not necessarily of characteristic 00) in which ω X= nΩ 1𝒪 X\omega_X=\wedge^n\Omega^1\simeq \mathcal{O}_X and also H j(X,𝒪 X)=0H^j(X, \mathcal{O}_X)=0 for all 1jn11\leq j \leq n-1.

If the base field is \mathbb{C}, then one can form the analyticification of XX and obtain a compact manifold that satisfies the first given definition.

Beware that there are slighlty different (and inequivalent) definitions in use. Notably in some contexts only the trivialization of the canonical bundle is required, but not the vanishing of the H 0<<n(X,𝒪 X)H^{0 \lt \bullet \lt n}(X,\mathcal{O}_X). To be explicit on this one sometimes speaks for emphasis of “strict” CY varieties when including this condition.

Examples

Properties

Artin-Mazur formal group

Over an algebraically closed field of positive characteristic an nn-dimensional Calabi-Yau variety XX has an Artin-Mazur formal group Φ X n\Phi^n_X which gives the deformation theory of the trivial line n-bundle over XX.

See also (Geer-Katsura 03).

As supersymmetric compactification spaces in string theory

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
Kähler manifoldU(k)2k2kKähler forms
Calabi-Yau manifoldSU(k)2k2k
hyper-Kähler manifoldSp(k)4k4k
G2 manifoldG277associative 3-form
Spin(7) manifoldSpin(7)8Cayley form

References

The original articles are

(…)

Surveys and reviews include

Discussion of the case of positive characteristic includes

The following page collects information on Calabi-Yau manifolds with an eye to application in string theory (e.g. supersymmetry and Calabi-Yau manifolds):

Discussion of the relation between the various shades of definitions includes

Revised on July 14, 2014 03:40:38 by Urs Schreiber (89.204.135.179)