Kähler manifold

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They are named after German mathematician Erich Kähler.



Symplectic viewpoint[edit]

A Kähler manifold is a symplectic manifold  (K,\omega) equipped with an integrable almost-complex structure which is compatible with the symplectic form.[1]

Complex viewpoint[edit]


Equivalence of definitions[edit]

Every Hermitian manifold  K is a complex manifold which comes naturally equipped with a Hermitian form  h and an integrable, almost complex structure  J . Assuming that  h is closed, there is a canonical symplectic form defined as  \omega = \frac i2 (h - \bar h ) which is compatible with  J , hence satisfying the first definition.

On the other hand, any symplectic form compatible with an almost complex structure must be a complex differential form of type  (1,1) , written in a coordinate chart  (U, z_i) as

 \omega = \frac i2 \sum_{j,k} h_{jk} dz_j \wedge d\bar{z_k}

for  h_{jk} \in C^\infty(U,\mathbb C) . The added assertions that  \omega be real-valued, closed, and non-degenerate guarantee that the  h_{jk} define Hermitian forms at each point in  K .[1]

Connection between Hermitian and symplectic definitions[edit]

Let  h be the Hermitian form,  \omega the symplectic form, and  J the almost complex structure. Since  \omega and  J are compatible, the new form  g(u,v) = \omega(u,Jv) is Riemannian.[1] One may then summarize the connection between these structures via the identity h=g + i\omega.

Kähler potentials[edit]

If  K is a complex manifold, it can be shown[1] that every strictly plurisubharmonic function  \rho \in C^\infty(K; \mathbb R) gives rise to a Kähler form as

 \omega = \frac i2 \partial \bar\partial \rho

where  \partial, \bar\partial are the Dolbeault operators. The function  \rho is said to be a Kähler potential.

In fact, utilizing the holomorphic version of the Poincaré lemma, a partial converse holds true locally. More specifically, if  (K,\omega) is a Kähler manifold then about every point  p \in K there is a neighbourhood  U containing  p and a function  \rho \in C^\infty(U,\mathbb R) such that  \omega\vert_U = i \partial \bar\partial \rho and here  \rho is termed a (local) Kähler potential.

Ricci tensor and Kähler manifolds[edit]

see Kähler manifolds in Ricci tensor.

The Laplacians on Kähler manifolds[edit]

On any Riemannian manifold we can define the Laplacian as \Delta_d=dd^*+d^*d where d is the exterior derivative and d^*=-(-1)^{nk}\star d\star, with the Hodge operator \star (alternatively, d^* is the adjoint of d with respect to the L^2 scalar product). Furthermore if X is Kähler then d and d^* are decomposed as

d=\partial+\bar{\partial},\ \ \ \ d^*=\partial^*+\bar{\partial}^*

and we can define two other Laplacians

\Delta_{\bar{\partial}}=\bar{\partial}\bar{\partial}^*+\bar{\partial}^*\bar{\partial},\ \ \ \ \Delta_\partial=\partial\partial^*+\partial^*\partial

that satisfy

\Delta_d=2\Delta_{\bar{\partial}}=2\Delta_\partial .



where \mathbf{H^r} is r-degree harmonic form and \mathbf{H}^{p,q} is {p,q}-degree harmonic form on X. Namely, a differential form \alpha is harmonic if and only if each \alpha^{i,j} belong to the {i,j}-degree harmonic form.

Further, if X is compact then we obtain

H^p(X,\Omega^q)\simeq H^{p,q}_{\bar{\partial}}(X)\simeq\mathbf{H}^{p,q}

where H^{p,q}_{\bar{\partial}}(X) is \bar{\partial}-harmonic cohomology group. This means that if \alpha is a differential form with {p,q}-degree there is only one element in {p,q}-harmonic form due to Dolbeault theorem.

Let h^{p,q}=\text{dim} H^{p,q}, called Hodge number, then we obtain

b_r=\sum_{p+q=r}h^{p,q},\ \ \ \ h^{p,q}=h^{q,p},\ \ \ \ h^{p,q}=h^{n-p,n-q}.

The LHS of the first identity, br, is r-th Betti number, the second identity comes from that since the Laplacian \Delta_d is a real operator H^{p,q}=\overline{H^{q,p}} and the third identity comes from Serre duality.


On a Kähler manifold, the associated Kähler form and metric are called Kähler–Einstein (or sometimes Einstein–Kähler) if its Ricci tensor is proportional to the metric tensor, R = \lambda g, for some constant λ. This name is a reminder of Einstein's considerations about the cosmological constant. See the article on Einstein manifolds for more details.

Originally the Kähler condition is independent on the Einstein condition, in which Ricci tensor is proportional to Riemannian metric with constant real number. The important point is that if X is Kähler then Christoffel symbols \Gamma^\alpha_{\beta\gamma} vanish and Ricci curvature is much simplified. The Kähler condition, therefore, is closely related with Ricci curvature. In fact Aubin and Yau proved the Calabi conjecture using the fact that on a compact Kähler manifold with the first Chern class c1=0 there is a unique Ricci-flat Kähler metric in each Kähler class. But in non-compact case the situation turns to be more complicated and the final solution might not be reached.



An important subclass of Kähler manifolds are Calabi–Yau manifolds.

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