Contents

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?) states any of the above conditions implies 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 simply connected 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 slightly 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

In terms of $G$-structure

Calabi-Yau structure is equivalently integrable G-structure for $G =$ SU(n).

Details are in Prins 16, Prop. 1.3.2. See also Vezzoni 06, p. 24.

In SU-bordism theory

We discuss the classes of Calabi-Yau manifolds seen in SU-bordism theory. For more see Calabi-Yau manifolds in SU-bordism theory.

Proposition

(K3-surface spans SU-bordism ring in degree 4)

The degree-4 generator $y_4 \in \Omega^{SU}_4$ in the SU-bordism ring (Prop. ) is represented by minus the class of any (non-torus) K3-surface:

$\Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle \,.$
Proposition

(Calabi-Yau manifolds generate the SU-bordism ring away from 2)

The SU-bordism ring away from 2 is multiplicatively generated by Calabi-Yau manifolds.

Proposition

(Calabi-Yau manifolds in complex dim $\leq 4$ generate the SU-bordism ring in $deg \leq 8$ away from 2)

There are Calabi-Yau manifolds of complex dimension $3$ and $4$ whose whose SU-bordism classes equal the generators $\pm y_6$ and $\pm y_8$ in Prop. .

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-structure$\,$$\,$special holonomy$\,$$\,$dimension$\,$$\,$preserved differential form$\,$
$\,\mathbb{C}\,$$\,$Kähler manifold$\,$$\,$U(n)$\,$$\,2n\,$$\,$Kähler forms $\omega_2\,$
$\,$Calabi-Yau manifold$\,$$\,$SU(n)$\,$$\,2n\,$
$\,\mathbb{H}\,$$\,$quaternionic Kähler manifold$\,$$\,$Sp(n).Sp(1)$\,$$\,4n\,$$\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$
$\,$hyper-Kähler manifold$\,$$\,$Sp(n)$\,$$\,4n\,$$\,\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\,$$\,$associative 3-form$\,$

References

The original articles are

(…)

Surveys and review:

Motivated from the relation between supersymmetry and Calabi-Yau manifolds:

In terms of G-structure:

• Luigi Vezzoni, The geometry of some special $SU(n)$-structure, 2006 (pdf, pdf)

• Daniël Prins, Section 1.3 of: On flux vacua, $SU(n)$-structures and generalised complex geometry, Université Claude Bernard – Lyon I, 2015. (arXiv:1602.05415, tel:01280717)

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

Discussion of Calabi-Yau orbifolds:

and in view of mirror symmetry:

• Shi-Shyr Roan, The mirror of Calabi-Yau orbifold, International Journal of Mathematics Vol. 02, No. 04, pp. 439-455 (1991) (doi:10.1142/S0129167X91000259)

• Alan Stapledon, New mirror pairs of Calabi-Yau orbifolds, Adv. Math. 230 (2012), no. 4-6, 1557-1596 (arXiv:1011.5006)

Last revised on January 10, 2023 at 22:36:21. See the history of this page for a list of all contributions to it.