Calabi-Yau Manifolds - Examples and Constructions

Constructing explicit examples of Calabi-Yau manifolds requires sophisticated techniques from algebraic geometry. Several systematic construction methods have been developed, each producing families of Calabi-Yau manifolds with different properties.

Hypersurfaces in Toric Varieties

The most systematic construction uses toric geometry. A Calabi-Yau hypersurface $X$ in a toric variety $V$ is given by vanishing of a section of an anticanonical line bundle:

$X = \{p \in V : s(p) = 0\}$

where $s \in H^0(V, -K_V)$ . The adjunction formula ensures $K_X \cong \mathcal{O}_X$ .

ExampleComplete Intersection Calabi-Yau

A complete intersection Calabi-Yau (CICY) is defined by multiple polynomial equations in a product of projective spaces: $X = \{(x,y,\ldots) \in \mathbb{P}^{n_1} \times \mathbb{P}^{n_2} \times \cdots : f_1 = \cdots = f_k = 0\}$

The configuration matrix encodes the multi-degrees. For example, the bi-cubic in $\mathbb{P}^2 \times \mathbb{P}^2$ defined by two degree $(3,3)$ polynomials is a Calabi-Yau threefold.

The classification of CICYs in products of projective spaces yields 7,890 distinct topological types of Calabi-Yau threefolds. Each configuration can be analyzed using linear algebra on the configuration matrix, making this a computationally tractable approach.

Elliptic Fibrations

Many Calabi-Yau threefolds admit an elliptic fibration structure $\pi: X \to B$ where $B$ is a complex surface and generic fibers are elliptic curves. The discriminant locus $\Delta \subset B$ parametrizes singular fibers.

DefinitionWeierstrass Model

An elliptic Calabi-Yau threefold in Weierstrass form over a base $B$ is given by: $y^2 = x^3 + fx + g$

where $f \in H^0(B, K_B^{-4})$ and $g \in H^0(B, K_B^{-6})$ . The discriminant $\Delta = 4f^3 + 27g^2$ must be a section of $K_B^{-12}$ .

For $X$ to be Calabi-Yau, we require $c_1(B) = c_1(K_B^{-1})$ to equal the class of $\Delta$ , which forces $B$ to have $c_1(B) = 12c_1(\mathcal{O}_B(1))$ for some line bundle. This is satisfied when $B = \mathbb{P}^2$ or $B$ is a Hirzebruch surface.

Quotient Constructions

Given a Calabi-Yau manifold $\tilde{X}$ and a finite group $G$ acting freely, the quotient $X = \tilde{X}/G$ is again Calabi-Yau. This provides examples with different topological invariants.

ExampleOrbifold Resolution

Starting with a torus quotient $\mathbb{T}^6/G$ where $G$ is a finite subgroup of $\text{SU}(3)$ , we obtain a singular space. Resolving the singularities using blow-ups produces a smooth Calabi-Yau threefold. Different resolution choices can yield distinct smooth manifolds.

Free Quotients and Fiber Products

The fiber product construction combines two Calabi-Yau manifolds. For elliptic Calabi-Yau threefolds $X_1 \to B$ and $X_2 \to B$ over the same base, the fiber product $X_1 \times_B X_2$ yields a Calabi-Yau fivefold.

Remark

Mirror symmetry predicts that many Calabi-Yau threefolds come in pairs $(X, X^\vee)$ with swapped Hodge numbers: $h^{1,1}(X) = h^{2,1}(X^\vee)$ and $h^{2,1}(X) = h^{1,1}(X^\vee)$ . Constructing both members of a mirror pair remains a major challenge, though toric methods have been successful in many cases.

The variety of construction methods reflects the richness of Calabi-Yau geometry and provides a diverse landscape for studying mirror symmetry and string compactifications.