Hodge Theory and Periods - Core Definitions

Hodge theory provides the fundamental framework for understanding the structure of cohomology on complex manifolds. For Calabi-Yau manifolds, the Hodge structure and associated period integrals encode essential geometric and topological information that plays a central role in mirror symmetry.

DefinitionHodge Decomposition

Let $X$ be a compact Kähler manifold. The Hodge decomposition states that the $k$ -th de Rham cohomology group decomposes as: $H^k(X,\mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(X)$

where $H^{p,q}(X) = H^q(X, \Omega^p_X)$ is the Dolbeault cohomology, consisting of $\bar{\partial}$ -closed $(p,q)$ -forms modulo $\bar{\partial}$ -exact forms.

This decomposition is compatible with complex conjugation: $\overline{H^{p,q}(X)} = H^{q,p}(X)$ . The Hodge numbers $h^{p,q}(X) = \dim H^{p,q}(X)$ determine the Hodge diamond, a visual representation of the cohomological structure.

For a Calabi-Yau threefold $X$ , the non-trivial Hodge numbers are organized as:

& & & 1 & & & \\ & & 0 & & 0 & & \\ & h^{2,1} & & h^{1,1} & & h^{2,1} & \\ 0 & & 0 & & 0 & & 0 \\ & h^{2,1} & & h^{1,1} & & h^{2,1} & \\ & & 0 & & 0 & & \\ & & & 1 & & & \end{array}$$ <Definition title="Period Integrals"> Let $X$ be a Calabi-Yau $n$-fold with holomorphic $n$-form $\Omega$. The **period integrals** are the integrals of $\Omega$ over a basis of $n$-cycles: $$\Pi_\gamma = \int_\gamma \Omega, \quad \gamma \in H_n(X,\mathbb{Z})$$ These complex numbers encode the complex structure of $X$ and satisfy the **Riemann bilinear relations**. </Definition> The periods are not independent; they satisfy differential equations as the complex structure varies. For a family of Calabi-Yau manifolds parametrized by a moduli space $\mathcal{M}$, the periods define a multi-valued holomorphic function on $\mathcal{M}$ called the **period map**. <Remark> The period map $\Phi: \mathcal{M} \to \mathcal{D}$ takes values in a period domain $\mathcal{D}$, which is a homogeneous space for a group related to the monodromy of the family. The image of the period map satisfies the Griffiths transversality condition, a fundamental constraint in Hodge theory. </Remark> ## Hodge Filtration The Hodge decomposition gives rise to the Hodge filtration on $H^n(X,\mathbb{C})$: $$F^p H^n(X,\mathbb{C}) = \bigoplus_{r \geq p} H^{r,n-r}(X)$$ This filtration satisfies: $$H^n(X,\mathbb{C}) = F^p \oplus \overline{F^{n-p+1}}$$ For Calabi-Yau threefolds, the middle cohomology $H^3(X,\mathbb{C})$ carries a pure Hodge structure of weight 3: $$H^3(X,\mathbb{C}) = H^{3,0} \oplus H^{2,1} \oplus H^{1,2} \oplus H^{0,3}$$ <Example title="Elliptic Curve Periods"> For an elliptic curve $E$ given by $y^2 = x(x-1)(x-\lambda)$, the period integral is: $$\omega(\lambda) = \int_{\gamma} \frac{dx}{y}$$ where $\gamma$ is a cycle in $H_1(E,\mathbb{Z})$. The ratio $\tau = \omega_2/\omega_1$ of two independent periods defines the modular parameter, which lies in the upper half-plane. </Example> The periods transform under monodromy around singular loci in moduli space, reflecting the topology of the family of Calabi-Yau manifolds. This monodromy action preserves the integral lattice structure and provides crucial arithmetic information.