complex geometry

# Contents

## Idea

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

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 $c_1(X) = 0$ 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.

## 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 $\omega_X=\wedge^n\Omega^1\simeq \mathcal{O}_X$ and also $H^j(X, \mathcal{O}_X)=0$ for all $1\leq j \leq n-1$.

If the base field is $\mathbb{C}$, then one can form the analyticification of $X$ 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 $H^{0 \lt \bullet \lt n}(X,\mathcal{O}_X)$. To be explicit on this one sometimes speaks for emphasis of “strict” CY varieties when including this condition.

## Properties

### Artin-Mazur formal group

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

### As supersymmetric compactification spaces in string theory

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
$\mathbb{C}$Kähler manifoldU(k)$2k$Kähler forms $\omega_2$
Calabi-Yau manifoldSU(k)$2k$
$\mathbb{H}$quaternionic Kähler manifoldSp(k)Sp(1)$4k$$\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$
hyper-Kähler manifoldSp(k)$4k$$\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2$ ($a^2 + b^2 + c^2 = 1$)
$\mathbb{O}$Spin(7) manifoldSpin(7)8Cayley form
G2 manifoldG2$7$associative 3-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

Mathematical review of the relation to quiver representations and mirror symmetry includes

• Yang-Hui He, Calabi-Yau Varieties: from Quiver Representations to Dessins d’Enfants (arXiv:1611.09398)

Discussion of CYs in positive characteristic includes

Revised on November 30, 2016 03:12:50 by Urs Schreiber (89.15.236.235)