Moduli Spaces - Main Theorem

The fundamental results about Calabi-Yau moduli spaces establish their smoothness, dimension, and geometric structure. These theorems provide the foundation for understanding deformations and constructing mirror pairs.

TheoremBogomolov-Tian-Todorov Theorem

Let $X$ be a Calabi-Yau manifold of dimension $n$ . Then the complex structure moduli space $\mathcal{M}_c$ is smooth at the point corresponding to $X$ , with dimension: $\dim\mathcal{M}_c = h^{n-1,1}(X)$

Moreover, the obstruction space $H^2(X,T_X)$ vanishes, so all infinitesimal deformations are unobstructed.

This remarkable result contrasts sharply with general varieties, where obstructions can prevent infinitesimal deformations from integrating to actual deformations. The proof uses the Calabi-Yau condition $K_X \cong \mathcal{O}_X$ in an essential way.

Remark

The BTT theorem applies to complex structure deformations. Kähler deformations are automatically unobstructed since $H^{1,1}(X,\mathbb{R})$ is a vector space with no obstructions. However, combined deformations mixing both types can have obstructions in some cases.

The smoothness of $\mathcal{M}_c$ at each point allows construction of a versal family. Near any Calabi-Yau threefold $X_0$ , there exists a family $\pi: \mathcal{X} \to B$ with fiber $X_0$ over $0 \in B$ , such that $B \cong \mathbb{C}^{h^{2,1}}$ and every nearby Calabi-Yau appears as a fiber.

TheoremLocal Torelli Theorem

For a Calabi-Yau threefold $X$ , the period map: $\Phi: \mathcal{M}_c \to \mathcal{D}/\Gamma$

from the complex structure moduli space to the period domain is a local isomorphism. That is, the differential: $d\Phi: T_{[X]}\mathcal{M}_c \to T_{\Phi([X])}\mathcal{D}/\Gamma$

is an isomorphism.

This result shows that infinitesimal complex structure deformations are completely determined by their effect on period integrals. The Kodaira-Spencer class of a deformation equals the derivative of the period map, connecting deformation theory with Hodge theory.

For threefolds with $h^{2,1} = 1$ , the global Torelli problem becomes whether the period map is injective globally. This holds for many examples but can fail when the manifold has large automorphism groups.

TheoremSpecial Kähler Structure

The complex structure moduli space $\mathcal{M}_c$ carries a natural special Kähler metric (the Weil-Petersson metric) with Kähler potential: $K = -\log\left(i\int_X \Omega \wedge \bar{\Omega}\right)$

The metric is: $g_{i\bar{j}} = \partial_i\partial_{\bar{j}}K$

This metric is Ricci-flat and has holonomy contained in $\text{U}(h^{2,1})$ .

The special Kähler structure encodes the variation of Hodge structure. The connection is flat in special coordinates (related to periods), and the curvature is controlled by the Yukawa coupling.

ExampleOne-Parameter Case

For a one-parameter family with coordinate $t$ , the Weil-Petersson metric is: $ds^2 = \frac{(\partial_t\omega)(\partial_{\bar{t}}\bar{\omega})}{(\omega\bar{\omega})} - \frac{(\partial_t\omega)(\omega\partial_{\bar{t}}\bar{\omega}) + (\bar{\omega}\partial_t\omega)(\partial_{\bar{t}}\bar{\omega})}{(\omega\bar{\omega})^2}$

where $\omega = \int_\gamma \Omega$ is a period. In the case of the mirror quintic, this can be computed explicitly from hypergeometric functions.

These structural theorems make Calabi-Yau moduli spaces tractable objects for both mathematical analysis and physical applications.