Hodge Theory and Periods - Main Theorem

The fundamental theorem in Hodge theory provides the cornerstone for understanding the cohomology of compact Kähler manifolds. This theorem has profound implications for Calabi-Yau geometry and mirror symmetry.

TheoremHodge Decomposition Theorem

Let $X$ be a compact Kähler manifold with Kähler form $\omega$ . Then:

The de Rham cohomology $H^k(X,\mathbb{C})$ decomposes as: $H^k(X,\mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(X)$ where $H^{p,q}(X) = \{\alpha \in \Omega^{p,q}(X) : \bar{\partial}\alpha = 0\}/\bar{\partial}\Omega^{p,q-1}(X)$
Every cohomology class in $H^{p,q}(X)$ has a unique harmonic representative $\alpha$ satisfying: $\Delta\alpha = 0, \quad \partial\alpha = \bar{\partial}\alpha = 0$ where $\Delta = \partial\partial^* + \partial^*\partial + \bar{\partial}\bar{\partial}^* + \bar{\partial}^*\bar{\partial}$ is the Hodge Laplacian
The space of harmonic $(p,q)$ -forms equals $H^{p,q}(X)$ : $\mathcal{H}^{p,q}(X) = H^{p,q}(X)$

The proof relies on Hodge theory for elliptic operators. The Hodge Laplacian $\Delta$ is an elliptic operator, and elliptic regularity implies that harmonic forms are smooth. The orthogonal projection onto the kernel of $\Delta$ provides the harmonic representative.

Remark

The Kähler condition is essential. For non-Kähler complex manifolds, the Hodge decomposition can fail. The classical example is the Hopf surface $S^1 \times S^3$ , which has $b_1 = 1$ but $h^{1,0} = 0$ , violating the Hodge decomposition $H^1 = H^{1,0} \oplus H^{0,1}$ .

Variation of Hodge Structure

For families of Calabi-Yau manifolds, the Hodge decomposition varies holomorphically, subject to important constraints.

TheoremGriffiths Transversality

Let $\pi: \mathcal{X} \to B$ be a smooth family of Calabi-Yau manifolds over a base $B$ . The Hodge filtration $F^\bullet$ on the cohomology bundle $\mathcal{H}^n = R^n\pi_*\mathbb{C}$ satisfies: $\nabla_{\text{GM}}(F^p) \subset \Omega^1_B \otimes F^{p-1}$

where $\nabla_{\text{GM}}$ is the Gauss-Manin connection. This condition is called Griffiths transversality.

Griffiths transversality constrains how the Hodge filtration can vary. It implies that the period map from the base $B$ to the classifying space of Hodge structures has restricted image. The infinitesimal period map is subject to these transversality constraints.

Torelli Theorems

A Torelli theorem states that a variety is determined by its Hodge structure. For Calabi-Yau manifolds, we have:

TheoremLocal Torelli for Calabi-Yau Threefolds

Let $X$ be a Calabi-Yau threefold. The infinitesimal period map: $\rho: T_{[X]}\mathcal{M}_c \to \text{Hom}(H^{3,0}(X), H^{2,1}(X))$

is an isomorphism. Equivalently, the Kodaira-Spencer map equals the differential of the period map.

This theorem ensures that first-order deformations of complex structure are detected by period integrals. For generic Calabi-Yau threefolds, the global Torelli theorem holds: the manifold is determined up to isomorphism by its periods.

Hodge Conjecture

While not directly about periods, the Hodge conjecture is intimately related to the Hodge structure on Calabi-Yau manifolds.

TheoremHodge Conjecture (Unproven)

Let $X$ be a non-singular projective variety over $\mathbb{C}$ . Then every Hodge class on $X$ is an integral linear combination of classes of algebraic cycles: $H^{2p}(X,\mathbb{Q}) \cap H^{p,p}(X) = \text{span}_\mathbb{Q}\{\text{classes of codimension-}p\text{ algebraic cycles}\}$

For Calabi-Yau threefolds, the Hodge conjecture would imply strong constraints on the geometry. The conjecture is known for $p=1$ (Lefschetz theorem on $(1,1)$ -classes) but remains open in general. It is one of the Clay Millennium Prize problems.

ExampleApplication to Mirror Symmetry

In mirror symmetry, the Hodge structure on the A-model side (counting curves) relates to the Hodge structure on the B-model side (periods). The mirror conjecture can be formulated in terms of Hodge-theoretic identifications, making Torelli-type theorems essential for establishing mirror correspondence.