The B-Model and Variations of Hodge Structure - Examples and Constructions

Explicit constructions of variations of Hodge structure illustrate the abstract theory and provide computational tools. Classical examples from algebraic geometry demonstrate the power of period methods.

ExampleElliptic Curves and Modular Forms

For the family of elliptic curves $E_\lambda: y^2 = x(x-1)(x-\lambda)$ , the period integral is: $\omega(\lambda) = \int_0^1 \frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$

This satisfies the Picard-Fuchs equation: $\lambda(1-\lambda)\frac{d^2\omega}{d\lambda^2} + (1-2\lambda)\frac{d\omega}{d\lambda} - \frac{1}{4}\omega = 0$

The solution is a hypergeometric function $_2F_1(\frac{1}{2}, \frac{1}{2}; 1; \lambda)$ , and the period map relates to the modular $j$ -invariant.

This example illustrates all key features: differential equations for periods, monodromy around singular points $\{0, 1, \infty\}$ , and connection to automorphic forms.

Mirror Quintic Variation

The mirror quintic family provides the prototypical Calabi-Yau example.

ExampleMirror Quintic VHS

The mirror quintic family has one complex structure parameter $\psi$ (or $z = \psi^{-5}$ ). The Hodge structure on $H^3$ has: $h^{3,0} = h^{0,3} = 1, \quad h^{2,1} = h^{1,2} = 1$

The period vector is $\Pi = (\Pi_0, \Pi_1, \partial_z\Pi_1, \partial_z^2\Pi_1)^T$ where: $\Pi_0(z) = \sum_{n=0}^\infty \frac{(5n)!}{(n!)^5} z^n$

is the fundamental period near $z=0$ .

The Picard-Fuchs operator is $\mathcal{L} = \theta^4 - 5z(5\theta+1)(5\theta+2)(5\theta+3)(5\theta+4)$ where $\theta = z\partial_z$ . This fourth-order equation reflects $\dim H^3 = 4$ .

K3 Surface Periods

K3 surfaces provide examples with large Hodge structures.

DefinitionK3 Period Domain

The period domain for K3 surfaces is: $\mathcal{D}_{K3} = \{\ell \in \mathbb{P}(H^2(S,\mathbb{C})) : (\ell, \ell) = 0, \, (\ell, \bar{\ell}) > 0\} / \text{Aut}(L_{K3})$

where $L_{K3}$ is the K3 lattice with signature $(3, 19)$ .

The period map $\mathcal{M}_{K3} \to \mathcal{D}_{K3}$ is surjective with dense image, and the Torelli theorem states it's injective up to automorphisms.

Toric Constructions

Toric geometry provides systematic methods for constructing variations via GKZ hypergeometric systems.

DefinitionGKZ System

For a toric Calabi-Yau with data encoded in matrix $A$ and parameter $\beta$ , periods satisfy: $\Box_i \Pi = 0, \quad Z_j \Pi = \beta_j \Pi$

where $\Box_i$ are toric differential operators and $Z_j$ are Euler operators. These form the GKZ hypergeometric system.

Solutions are expressible as series: $\Pi_\gamma(z) = \sum_{m \in L_A} \frac{\Gamma(\langle\gamma, m\rangle)}{\Gamma(Am + \beta)} z^m$

where $L_A$ is the lattice associated to $A$ .

Deformation to the Normal Cone

Deformation to the normal cone provides geometric understanding of limiting processes.

Remark

When a Calabi-Yau $X$ degenerates to $X_0$ along a divisor $D$ , the variation of Hodge structure extends to the boundary with logarithmic singularities. The residue map: $\text{Res}: H^n(X) \to H^{n-1}(D)$

relates the VHS on $X$ to that on $D$ , explaining the monodromy weight filtration geometrically.

Computational Techniques

Computing period integrals requires combining analytic and algebraic methods.

ExampleSeries Solutions

Near large complex structure limit $z \to 0$ , periods expand as: $\Pi(z) = \sum_{n=0}^\infty a_n z^n(\log z)^k$

Coefficients are determined recursively from the Picard-Fuchs equation. For the quintic: $a_n = \frac{(5n)!}{(n!)^5}$

which grows factorially, reflecting convergence radius zero.

These examples demonstrate how abstract VHS theory becomes concrete through explicit calculations, providing bridges between algebraic geometry, differential equations, and special functions.