geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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.
The language used in this article is implicitly analytic, rather than algebraic. Is this OK? Or should I make this explicit?
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.
in dimension 1: elliptic curve
In dimension 2: K3 surface.
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 $X$.
See also (Geer-Katsura 03).
classification of special holonomy manifolds by Berger's theorem:
G-structure | special holonomy | dimension | preserved differential form | |
---|---|---|---|---|
$\mathbb{C}$ | Kähler manifold | U(k) | $2k$ | Kähler forms $\omega_2$ |
Calabi-Yau manifold | SU(k) | $2k$ | ||
$\mathbb{H}$ | quaternionic Kähler manifold | Sp(k)Sp(1) | $4k$ | $\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$ |
hyper-Kähler manifold | Sp(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) manifold | Spin(7) | 8 | Cayley form |
G2 manifold | G2 | $7$ | associative 3-form |
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
Discussion of CYs in positive characteristic includes
Philip Candelas, Xenia de la Ossa, Fernando Rodriguez-Villegas, Calabi-Yau Manifolds Over Finite Fields, I (arXiv:hep-th/0012233)
Philip Candelas, Xenia de la Ossa, Fernando Rodriguez-Villegas, Calabi-Yau Manifolds Over Finite Fields, II (arXiv:hep-th/0402133)