Calabi-Yau Manifolds - Main Theorem

The foundational result in Calabi-Yau geometry is the Calabi conjecture, proved by Yau in 1977. This theorem bridges the gap between topological and analytic conditions, establishing the existence of Ricci-flat metrics on manifolds with vanishing first Chern class.

TheoremCalabi-Yau Theorem (Yau, 1977)

Let $X$ be a compact Kähler manifold with $c_1(X) = 0$ . Then for each Kähler class $[\omega] \in H^{1,1}(X,\mathbb{R})$ , there exists a unique Ricci-flat Kähler metric $g$ with Kähler form in the class $[\omega]$ .

Moreover, if $X$ admits a nowhere-vanishing holomorphic $(n,0)$ -form $\Omega$ , then $g$ is the unique metric satisfying: $\text{Ric}(g) = 0, \quad \omega^n = c \cdot \Omega \wedge \bar{\Omega}$ for some constant $c > 0$ .

This theorem is the culmination of Calabi's conjecture from 1954, which proposed that the vanishing of the first Chern class should guarantee the existence of a Kähler metric with vanishing Ricci curvature. The proof employs sophisticated techniques from nonlinear partial differential equations, specifically the complex Monge-Ampère equation.

The Monge-Ampère Equation

The key to proving the theorem is solving the complex Monge-Ampère equation. Given a reference Kähler metric $\omega_0$ in the class $[\omega]$ , we seek a function $\varphi: X \to \mathbb{R}$ such that the modified Kähler form $\omega = \omega_0 + i\partial\bar{\partial}\varphi$ satisfies:

$\frac{\omega^n}{\omega_0^n} = e^{F}$

where $F$ is determined by the condition $c_1(X) = 0$ . The Ricci-flatness condition translates to: $\text{Ric}(\omega) = -i\partial\bar{\partial}\log(\omega^n) = 0$

Remark

The uniqueness part of the theorem is crucial for applications. It implies that the Ricci-flat metric is completely determined by its Kähler class and the choice of holomorphic volume form, providing a canonical metric structure on Calabi-Yau manifolds.

Implications for Holonomy

An immediate corollary of Yau's theorem concerns the holonomy group. Since the metric is Ricci-flat and Kähler, Berger's classification of holonomy groups implies:

TheoremHolonomy Reduction

Let $(X,g)$ be a compact Ricci-flat Kähler $n$ -fold with $c_1(X) = 0$ . If $X$ is simply connected and irreducible, then: $\text{Hol}(g) = \text{SU}(n)$

This means the holonomy group is exactly $\text{SU}(n)$ , not just contained in it.

The reduction from $\text{U}(n)$ to $\text{SU}(n)$ reflects the triviality of the canonical bundle. The Kähler form $\omega$ is parallel, accounting for the $\text{U}(1)$ factor in $\text{U}(n) = (\text{SU}(n) \times \text{U}(1))/\mathbb{Z}_n$ , while the vanishing of $c_1$ removes this factor from the holonomy.

Deformation Theory

The theorem also has profound consequences for deformation theory. The space of Ricci-flat metrics modulo diffeomorphisms is described by:

$\mathcal{M}_{\text{RF}} \cong \mathcal{M}_K \times \mathcal{M}_c$

where $\mathcal{M}_K$ is the Kähler moduli space and $\mathcal{M}_c$ is the complex structure moduli space. This local product structure near any point in moduli space reflects the splitting of deformations into Kähler and complex structure variations.

ExampleUniqueness Application

For the quintic threefold $X_5 \subset \mathbb{P}^4$ , there is a unique Ricci-flat metric in the hyperplane class up to scaling. This metric cannot be written in closed form but its existence and uniqueness are guaranteed by Yau's theorem.

The Calabi-Yau theorem provides the geometric foundation for all subsequent developments in mirror symmetry, ensuring that the topological definition of Calabi-Yau manifolds corresponds to a rich analytic structure.