# Kähler manifold

In mathematics and especially differential geometry, a **Kähler manifold** is a manifold with three mutually compatible structures; a complex structure, a Riemannian structure, and a symplectic structure. On a Kähler manifold X there exists Kähler potential and the Levi-Civita connection corresponding to the metric of X gives rise to a connection on the canonical line bundle.

Smooth projective algebraic varieties are examples of Kähler manifolds. By Kodaira embedding theorem, Kähler manifolds that have a positive line bundle can always be embedded into projective spaces.

They are named after German mathematician Erich Kähler.

## Definitions

Since Kähler manifolds are naturally equipped with several compatible structures, there are many equivalent ways of creating Kähler forms.

### Symplectic viewpoint

A Kähler manifold is a symplectic manifold equipped with an integrable almost-complex structure which is compatible with the symplectic form.^{[1]}

### Complex viewpoint

A Kähler manifold is a Hermitian manifold whose associated Hermitian form is closed. The closed Hermitian form is called the **Kähler metric**.

### Equivalence of definitions

Every Hermitian manifold is a complex manifold which comes naturally equipped with a Hermitian form and an integrable, almost complex structure . Assuming that is closed, there is a canonical symplectic form defined as which is compatible with , 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 , written in a coordinate chart as

for . The added assertions that be real-valued, closed, and non-degenerate guarantee that the define Hermitian forms at each point in .^{[1]}

## Connection between Hermitian and symplectic definitions

Let be the Hermitian form, the symplectic form, and the almost complex structure. Since and are compatible, the new form is Riemannian.^{[1]} One may then summarize the connection between these structures via the identity .

## Kähler potentials

If is a complex manifold, it can be shown^{[1]} that every strictly plurisubharmonic function gives rise to a Kähler form as

where are the Dolbeault operators. The function 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 is a Kähler manifold then about every point there is a neighbourhood containing and a function such that and here is termed a **(local) Kähler potential.**

## Ricci tensor and Kähler manifolds

## The Laplacians on Kähler manifolds

Let be the Hodge operator and then on a differential manifold *X* we can define the Laplacian as
where is the exterior derivative and . Furthermore if *X* is Kähler then and are decomposed as

and we can define another Laplacians

that satisfy

From these facts we obtain the Hodge decomposition (see Hodge theory)

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

Further, if *X* is compact then we obtain

where is -harmonic cohomology group. This means that if is a differential form with {p,q}-degree there is only one element in {p,q}-harmonic form due to Dolbeault theorem.

Let , called Hodge number, then we obtain

The LHS of the first identity, *b _{r}*, is r-th Betti number, the second identity comes from that since the Laplacian is a real operator and the third identity comes from Serre duality.

## Applications

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, , 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 vanish and Ricci curvature is much simplified. The Kähler condition, therefore, is closely related with Ricci curvature. In fact Aubin and Yau prove the Calabi conjecture using the fact that on a compact Kähler manifold with the first Chern class *c _{1}=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.

## Examples

- Complex Euclidean space
**C**^{n}with the standard Hermitian metric is a Kähler manifold. - A torus
**C**^{n}/Λ (Λ a full lattice) inherits a flat metric from the Euclidean metric on**C**^{n}, and is therefore a compact Kähler manifold. - Every Riemannian metric on a Riemann surface is Kähler, since the condition for
*ω*to be closed is trivial in 2 (real) dimensions. - Complex projective space
**CP**^{n}admits a homogeneous Kähler metric, the Fubini–Study metric. An Hermitian form in (the vector space)**C**^{n + 1}defines a unitary subgroup*U*(*n*+ 1) in*GL*(*n*+ 1,*C*); a Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a*U*(*n*+ 1) action. By elementary linear algebra, any two Fubini–Study metrics are isometric under a projective automorphism of**CP**^{n}, so it is common to speak of "the" Fubini–Study metric. - The induced metric on a complex submanifold of a Kähler manifold is Kähler. In particular, any Stein manifold (embedded in
**C**^{n}) or projective algebraic variety (embedded in**CP**^{n}) is of Kähler type. This is fundamental to their analytic theory. - The unit complex ball
**B**^{n}admits a Kähler metric called the Bergman metric which has constant holomorphic sectional curvature. - Every K3 surface is Kähler (by a theorem of Y.-T. Siu).

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

## See also

- Hermitian manifold
- Almost complex manifold
- Hyper-Kähler manifold
- Kähler–Einstein metric
- Quaternion-Kähler manifold
- Complex Poisson manifold
- Einstein manifold
- Calabi conjecture

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}{{#invoke:citation/CS1|citation |CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- Alan Huckleberry and Tilman Wurzbacher, eds.
*Infinite Dimensional Kähler Manifolds*(2001), Birkhauser Verlag, Basel ISBN 3-7643-6602-8. - {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- Andrei Moroianu,
*Lectures on Kähler Geometry*(2004), http://www.math.polytechnique.fr/~moroianu/tex/kg.pdf - André Weil,
*Introduction à l'étude des variétés kählériennes*(1958)

## External links

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}