Kähler Manifolds and Calabi–Yau

A Kähler manifold is a complex manifold whose complex structure, Riemannian metric and symplectic form are not three pieces of data but one, in three guises. That compatibility is the Kähler condition, and it is the reason complex geometry has a Hodge theory at all: on a Kähler manifold the Hodge Laplacian splits into pieces that respect the complex structure, so the cohomology computed in Hodge theory acquires a bigrading and the Betti numbers refine into Hodge numbers.

The destination of the page is Yau’s solution of the Calabi conjecture: a Kähler manifold admits a Ricci-flat Kähler metric in each class $c_1(M) = 0$, and such manifolds are Calabi–Yau. That theorem is what makes Calabi–Yau manifolds the standard answer to “which compact manifolds carry supersymmetry”, since Ricci-flatness is the vacuum Einstein condition and Kählerianity is what preserves the complex structure.

Complex projective space $\mathbb{CP}^n$ is the model. It has a complex structure (holomorphic coordinates), a metric (the Fubini–Study metric), and a symplectic form (the curvature form of the hyperplane bundle). The Kähler condition says these agree: the metric’s associated 2-form is closed, which is the symplectic condition, and it equals the Hermitian form, which is the complex condition.

The reason Kähler manifolds are ubiquitous is that any complex submanifold of $\mathbb{CP}^n$ is Kähler, by restricting the Fubini–Study metric. So every smooth projective variety is Kähler, and the theory applies to the whole of algebraic geometry. That is why a condition that looks restrictive is in practice the standard setting.

Definition. A complex structure on a real $2n$-manifold $M$ is a bundle endomorphism $J : TM \to TM$ with $J^2 = -\mathrm{id}$. The metric $g$ is Hermitian if $g(Jv, Jw) = g(v,w)$, and then

is the associated 2-form, called the Kähler form when it is closed.

Definition. $(M, J, g, \omega)$ is Kähler if $\mathrm{d}\omega = 0$.

That is the whole definition, and it is deceptively thin: a single closedness condition. Its content is in the equivalences below.

Proposition (three forms of the Kähler condition). The following are equivalent:

1. $\mathrm{d}\omega = 0$ (the symplectic form is closed);
2. $\nabla J = 0$, where $\nabla$ is the Levi-Civita connection of $g$ (the complex structure is parallel);
3. around every point there are holomorphic coordinates in which the metric’s coefficients agree to first order with the constant metric.

Proof sketch. (1) $\Leftrightarrow$ (2) is the identity $2\,g((\nabla_X J)Y, Z) = \mathrm{d}\omega(JX, Y, Z) + \mathrm{d}\omega(X, JY, Z) - \mathrm{d}\omega(X, Y, JZ)$, verified by expanding both sides with $\nabla g = 0$ and $J^2 = -1$: the symplectic form’s three arguments can each be rotated by $J$, and the closure of $\omega$ kills the symmetric part while $\nabla J = 0$ kills it directly.

(2) $\Rightarrow$ (3): if $\nabla J = 0$ then at any point $p$ normal coordinates can be chosen for $g$ that are also holomorphic, because the complex structure is constant to first order at $p$; conversely holomorphic normal coordinates give $\partial_k g_{i\bar\jmath}(p) = 0$ for all indices, which is $\nabla J(p) = 0$. $\blacksquare$

The middle form is the one to remember: $J$ is parallel. A Kähler manifold is one where the complex structure is not merely present but preserved by the geometry, and everything special about Kähler manifolds follows from that conservation law.

This is the technical heart of the subject, and the reason Hodge theory is better on Kähler manifolds than on general Riemannian ones. Complexify the forms and extend $J$; then $d$ splits into holomorphic and antiholomorphic pieces

where $\Omega^{p,q}$ is the space of forms with $p$ holomorphic and $q$ antiholomorphic indices. There are adjoints $\partial^*, \bar\partial^*$ and Laplacians $\Delta_d, \Delta_\partial, \Delta_{\bar\partial}$.

Theorem (Kähler identities). On a Kähler manifold,

where $\Lambda$ is the adjoint of wedging with the Kähler form $\omega \wedge -$.

Proof idea. The commutator identities follow from a representation-theoretic computation: wedging with $\omega$, contraction $\Lambda$, and the degree counting operator together generate a representation of the Lie algebra $\mathfrak{sl}_2$ on each fibre, and the identities are the bracket relations of that representation computed with $\mathrm{d}\omega = 0$; the closure of the Kähler form is exactly what kills the error terms. The Laplacian identity then follows by taking adjoints of the commutator identities and combining: $\Delta_d = (\partial + \bar\partial)(\partial^* + \bar\partial^*) + (\partial^* + \bar\partial^*)(\partial + \bar\partial)$ expands into eight terms, and the commutator identities pair four of them into $\Delta_\partial$ and four into $\Delta_{\bar\partial}$, each coming out equal to $\Delta_d$. The full expansion is a page of signs and is carried out in Huybrechts §3.1 or Griffiths–Harris pp. 111–115; the structure of the argument; $\mathfrak{sl}_2$ representation plus one closedness hypothesis; is the part to remember. $\blacksquare$

Corollary (the Kähler package). On a compact Kähler manifold:

so the Betti numbers refine into Hodge numbers $h^{p,q} = \dim\mathcal{H}^{p,q}$, and they satisfy $h^{p,q} = h^{q,p}$ and $h^{p,q} = h^{n-p,n-q}$.

Proof. $\Delta_d = 2\Delta_{\bar\partial}$ says a form is $d$-harmonic iff it is $\bar\partial$-harmonic, so the $d$-harmonic forms decompose according to the $(p,q)$-bigrading, which the $\bar\partial$-Laplacian respects. Complex conjugation exchanges $\partial$ and $\bar\partial$, giving the second identity; the Hodge star gives the third, as in the real case. $\blacksquare$

The $h^{p,q} = h^{q,p}$ symmetry is why the Hodge diamond is drawn as a diamond, and it is a genuine restriction: the Betti numbers are not arbitrary, since $b_k = \sum_{p+q=k} h^{p,q}$ with the pairs forced equal.

The bridge to Calabi–Yau is the Ricci form. On a Kähler manifold, in holomorphic coordinates,

is a closed $(1,1)$-form representing $2\pi c_1(M)$, the first Chern class of the canonical bundle $K_M = \Lambda^{n,0}T^*M$. Its relation to Ricci curvature is $\rho(X, Y) = \operatorname{Ric}(JX, Y)$.

Theorem (Yau’s resolution of the Calabi conjecture, 1976). Let $M$ be a compact Kähler manifold and $\omega$ a Kähler form. Then for any $(1,1)$-class $\alpha$ representing $2\pi c_1(M)$, there is a unique Kähler form $\omega' = \omega + i\partial\bar\partial\varphi$ in the same class whose Ricci form represents $\alpha$.

Proof sketch. The equation reduces to a complex Monge–Ampère equation for the potential $\varphi$:

with $F$ determined by $\alpha$. Existence is a fully nonlinear PDE theorem and is not proved here; the continuity method gives a priori estimates in $C^0$ and $C^2$ from the maximum principle and the comparison of volumes, and the $C^{2,\alpha}$ and smooth estimates close the loop. What makes the equation tractable is precisely the Kähler condition: the determinant on the left is the Jacobian of a map on forms in a fixed cohomology class, so the equation is a prescribed-volume-form problem, and Yau’s a priori estimates are uniform in the continuity parameter. Uniqueness follows from the maximum principle applied to the difference of two potentials. $\blacksquare$

The theorem is a black-box PDE result plus Kähler bookkeeping, and the page says so. What the bookkeeping contributes is real: without the Kähler condition the equation has no cohomological obstruction to solve, and $c_1(M)$ is exactly what says whether a Ricci-flat metric is possible at all.

Definition. A Calabi–Yau manifold is a compact Kähler manifold with trivial canonical bundle; by Yau’s theorem it admits a unique Ricci-flat Kähler metric in each Kähler class.

• $\mathbb{CP}^n$. Kähler with the Fubini–Study metric, but $c_1(\mathbb{CP}^n) = (n+1)[H] \neq 0$, so never Calabi–Yau for $n \geq 1$. Positive Ricci curvature, consistent with it being Fano.
• Complex tori $\mathbb{C}^n/\Lambda$. Kähler with the flat metric; $c_1 = 0$ and the flat metric is Ricci-flat. Calabi–Yau in the differential- geometric sense, though the holonomy is trivial rather than $SU(n)$, so algebraic geometers often reserve the name for the simply connected case.
• Quintic threefold in $\mathbb{CP}^4$. By adjunction $K_X = K_{\mathbb{CP}^4} \otimes \mathcal{O}(5)|_X = \mathcal{O}(5-5)|_X$ is trivial, so $c_1 = 0$: the quintic is Calabi–Yau, and it is the standard example in string theory. Yau’s theorem produces the Ricci-flat metric, which is never known explicitly; the theorem proves existence, and in applications the metric is approximated numerically.

Supersymmetric compactification wants the extra dimensions to carry a geometry preserving some supersymmetry, which forces the holonomy to be contained in $SU(n)$; and on a Kähler manifold, holonomy contained in $SU(n)$ is equivalent to Ricci-flatness. So the string-theory requirement is exactly Yau’s equation, and the moduli of Ricci-flat metrics (Hodge-theoretic, by the Bogomolov–Tian– Todorov theorem) is what gives the effective four-dimensional theory its particle spectrum. The Hodge numbers $h^{1,1}$ and $h^{2,1}$ count Kähler and complex-structure moduli, and mirror symmetry; the reflection $h^{p,q}(M) = h^{n-p,q}(M^\vee)$; exchanges them.

• Assuming Kähler means “complex and symplectic”. A complex manifold with a symplectic form compatible with $J$ need not be Kähler: the Kähler form must be closed, and that is a genuine condition. The Hopf surface is complex and admits a compatible symplectic form but is not Kähler; its $b_1 = 1$, while Kähler manifolds have odd Betti numbers even.
• Quoting the Kähler identities on a Hermitian manifold. They fail without $\mathrm{d}\omega = 0$, and the failure is not a small correction: on a general Hermitian manifold $\Delta_d$ does not respect the $(p,q)$-bigrading at all, so the Hodge numbers do not exist.
• Reading $c_1 = 0$ as sufficient for Ricci-flatness without Yau. It is sufficient given Yau’s theorem, and the theorem needs the Kähler class to be fixed and the manifold compact. The existence of a Ricci-flat metric is a deep PDE result, not an algebraic identity.
• Confusing Calabi–Yau as differential geometer and algebraic geometer use it. The former means compact Ricci-flat Kähler; the latter usually adds simply connected and holonomy exactly $SU(n)$, which excludes tori. State which convention is in use.
• Expecting explicit Ricci-flat metrics. Yau’s theorem is an existence theorem. On a generic Calabi–Yau threefold no Ricci-flat metric is known in closed form, and numerical approximation is an active field.

• Hodge Theory; the real-case Hodge decomposition, here refined by the bigrading.
• De Rham Cohomology; the cohomology being refined into Hodge numbers.
• Curvature; Ricci curvature and the sectional curvature picture.
• Riemannian Geometry; the Levi-Civita connection whose parallelism of $J$ defines the condition.
• The Gauss–Bonnet Theorem; $\chi = \sum_{p,q}(-1)^{p+q}h^{p,q}$, the Hodge diamond’s alternating sum.
• Hodge Theory Applications in the Summary; where Calabi–Yau moduli enter the track’s summary table.
• Beyond the Standard Model; string compactification, the applied reason for the subject.

• Applications
• Curvature
• De Rham Cohomology
• Differential Forms