nLab Calabi-Yau variety

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.

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 $X$ 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 analytification 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.

Examples

• in dimension 1: an elliptic curve is a pointed Calabi-Yau $1$-fold.

• In dimension 2: K3 surface.

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:

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

(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 $\leq 4$ 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. .

(CLP 19, Theorem 13.5b,c)

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 $X$.

See also (Geer-Katsura 03).

As supersymmetric compactification spaces in string theory

• supersymmetry and Calabi-Yau manifolds

• heterotic string theory on CY3-manifolds

Related concepts

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$\,$

• Calabi-Yau object

  • Calabi-Yau algebra, Calabi-Yau manifold, generalized Calabi-Yau manifold

• Calabi-Yau cohomology

• moduli space of Calabi-Yau spaces

• SU-cobordism theory

References

The original articles:

• Shing-Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equation, Communications on Pure and Applied Mathematics,

  31 3 (1978) 339-411 [doi:10.1002/cpa.3160310304]

Surveys and review:

• Shing-Tung Yau, Calabi-Yau manifold, Scholarpedia 4 86524 (2009) [doi]

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

• Tristan Hübsch, Calabi-Yau Manifolds – A Bestiary for Physicists, World Scientific 1992 (doi:10.1142/1410)

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:[phone])

Discussion of the case of positive characteristic includes

• Gerard van der Geer, T. Katsura, On the height of Calabi-Yau varieties in positive characteristic (arXiv:math/0302023)

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

• Sheldon Katz, Rolf Schimmrigk, Andreas Wißkirchen, CALABI-YAU

Discussion of the relation between the various shades of definitions includes

• MathOverflow: Calabi-Yau manifolds

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

• 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)

Discussion of Calabi-Yau orbifolds:

• Dominic Joyce, On the topology of desingularizations of Calabi-Yau orbifolds (arXiv:math/9806146, spire:485280)

• Dominic Joyce, Deforming Calabi-Yau orbifolds, Asian Journal of Mathematics 3.4 (1999): 853-868 (doi:10.4310/AJM.1999.v3.n4.a7 pdf)

• Dominic Joyce, Section 6.5.1 of: Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press (2000) (ISBN:9780198506010)

• Wei-Dong Ruan, Yuguang Zhang, Convergence of Calabi-Yau manifolds, Advances in Mathematics Volume 228, Issue 3, 20 October 2011, Pages 1543-1589 (arXiv:0905.3424, doi:10.1016/j.aim.2011.06.023)

• Ronan J. Conlon, Anda Degeratu, Frédéric Rochon, Quasi-asymptotically conical Calabi-Yau manifolds. Geom. Topol. 23 (2019) 29-100 (arXiv:1611.04410)

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:

• Ivan Limonchenko, Zhi Lu, Taras Panov, Calabi-Yau hypersurfaces and SU-bordism, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 (arXiv:1712.07350)

• Georgy Chernykh, Ivan Limonchenko, Taras Panov, $SU$-bordism: structure results and geometric representatives, Russian Math. Surveys 74 (2019), no. 3, 461-524 (arXiv:1903.07178)

• Taras Panov, A geometric view on $SU$-bordism, talk at Moscow State University 2020 (webpage, pdf, pdf)