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 ${\displaystyle (K,\omega )}$ 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 ${\displaystyle K}$ is a complex manifold which comes naturally equipped with a Hermitian form ${\displaystyle h}$ and an integrable, almost complex structure ${\displaystyle J}$. Assuming that ${\displaystyle h}$ is closed, there is a canonical symplectic form defined as ${\displaystyle \omega ={\frac {i}{2}}(h-{\bar {h}})}$ which is compatible with ${\displaystyle J}$, 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 ${\displaystyle (1,1)}$, written in a coordinate chart ${\displaystyle (U,z_{i})}$ as

${\displaystyle \omega ={\frac {i}{2}}\sum _{j,k}h_{jk}dz_{j}\wedge d{\bar {z_{k}}}}$

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

Connection between Hermitian and symplectic definitions

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

Kähler potentials

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

${\displaystyle \omega ={\frac {i}{2}}\partial {\bar {\partial }}\rho }$

where ${\displaystyle \partial ,{\bar {\partial }}}$ are the Dolbeault operators. The function ${\displaystyle \rho }$ 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 ${\displaystyle (K,\omega )}$ is a Kähler manifold then about every point ${\displaystyle p\in K}$ there is a neighbourhood ${\displaystyle U}$ containing ${\displaystyle p}$ and a function ${\displaystyle \rho \in C^{\infty }(U,\mathbb {R} )}$ such that ${\displaystyle \omega \vert _{U}=i\partial {\bar {\partial }}\rho }$ and here ${\displaystyle \rho }$ is termed a (local) Kähler potential.

Ricci tensor and Kähler manifolds

see Kähler manifolds in Ricci tensor.

The Laplacians on Kähler manifolds

Let ${\displaystyle \star }$ be the Hodge operator and then on a differential manifold X we can define the Laplacian as ${\displaystyle \Delta _{d}=dd^{*}+d^{*}d}$ where ${\displaystyle d}$ is the exterior derivative and ${\displaystyle d^{*}=-(-1)^{nk}\star d\star }$. Furthermore if X is Kähler then ${\displaystyle d}$ and ${\displaystyle d^{*}}$ are decomposed as

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

and we can define another Laplacians

${\displaystyle \Delta _{\bar {\partial }}={\bar {\partial }}{\bar {\partial }}^{*}+{\bar {\partial }}^{*}{\bar {\partial }},\ \ \ \ \Delta _{\partial }=\partial \partial ^{*}+\partial ^{*}\partial }$

that satisfy

${\displaystyle \Delta _{d}=2\Delta _{\bar {\partial }}=2\Delta _{\partial }.}$

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

${\displaystyle \mathbf {H^{r}} =\bigoplus _{p+q=r}\mathbf {H} ^{p,q}}$

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

Further, if X is compact then we obtain

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

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

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

${\displaystyle b_{r}=\sum _{p+q=r}h^{p,q},\ \ \ \ h^{p,q}=h^{q,p},\ \ \ \ h^{p,q}=h^{n-p,n-q}.}$

The LHS of the first identity, br, is r-th Betti number, the second identity comes from that since the Laplacian ${\displaystyle \Delta _{d}}$ is a real operator ${\displaystyle H^{p,q}={\overline {H^{q,p}}}}$ 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, ${\displaystyle R=\lambda g}$, 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 ${\displaystyle \Gamma _{\beta \gamma }^{\alpha }}$ 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 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.

Examples

1. Complex Euclidean space Cn with the standard Hermitian metric is a Kähler manifold.
2. A torus Cn/Λ (Λ a full lattice) inherits a flat metric from the Euclidean metric on Cn, and is therefore a compact Kähler manifold.
3. Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 (real) dimensions.
4. Complex projective space CPn admits a homogeneous Kähler metric, the Fubini–Study metric. An Hermitian form in (the vector space) Cn + 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 CPn, so it is common to speak of "the" Fubini–Study metric.
5. The induced metric on a complex submanifold of a Kähler manifold is Kähler. In particular, any Stein manifold (embedded in Cn) or projective algebraic variety (embedded in CPn) is of Kähler type. This is fundamental to their analytic theory.
6. The unit complex ball Bn admits a Kähler metric called the Bergman metric which has constant holomorphic sectional curvature.
7. Every K3 surface is Kähler (by a theorem of Y.-T. Siu).

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

References

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• Alan Huckleberry and Tilman Wurzbacher, eds. Infinite Dimensional Kähler Manifolds (2001), Birkhauser Verlag, Basel ISBN 3-7643-6602-8.
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