nLab
Calabi-Yau variety

Contents

Contents

Idea

An nn-dimensional Calabi-Yau variety is an nn-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 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?) 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)=0c_1(X) = 0 implies in general that the canonical bundle is topologically trivial. But if XX is a simply connected 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 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<<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

In terms of GG-structure

Calabi-Yau structure is equivalently integrable G-structure for G=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 4Ω 4 SUy_4 \in \Omega^{SU}_4 in the SU-bordism ring (Prop. ) is represented by minus the class of any (non-torus) K3-surface:

Ω 4 SU[12][K3]. \Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle \,.

(LLP 17, Lemma 1.5, Example 3.1, CLP 19, Theorem 13.5a)

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.

(LLP 17, Theorem 2.4)

Proposition

(Calabi-Yau manifolds in complex dim 4\leq 4 generate the SU-bordism ring in deg8deg \leq 8 away from 2)

There are Calabi-Yau manifolds of complex dimension 33 and 44 whose whose SU-bordism classes equal the generators ±y 6\pm y_6 and ±y 8\pm y_8 in Prop. .

(CLP 19, Theorem 13.5b,c)

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-structure\,\,special holonomy\,\,dimension\,\,preserved differential form\,
\,\mathbb{C}\,\,Kähler manifold\,\,U(n)\,2n\,2n\,\,Kähler forms ω 2\omega_2\,
\,Calabi-Yau manifold\,\,SU(n)\,2n\,2n\,
\,\mathbb{H}\,\,quaternionic Kähler manifold\,\,Sp(n).Sp(1)\,4n\,4n\,ω 4=ω 1ω 1+ω 2ω 2+ω 3ω 3\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,
\,hyper-Kähler manifold\,\,Sp(n)\,4n\,4n\,ω=aω 2 (1)+bω 2 (2)+cω 2 (3)\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\, (a 2+b 2+c 2=1a^2 + b^2 + c^2 = 1)
𝕆\,\mathbb{O}\,\,Spin(7) manifold\,\,Spin(7)\,\,8\,\,Cayley form\,
\,G2 manifold\,\,G2\,7\,7\,\,associative 3-form\,

References

The original articles are

(…)

Surveys and reviews include

In terms of G-structure:

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

  • Daniël Prins, Section 1.3 of: On flux vacua, SU(n)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)

On Calabi-Yau manifolds in SU-bordism theory:

Last revised on November 20, 2020 at 14:29:35. See the history of this page for a list of all contributions to it.