User:Johannes nordstrom/sandbox

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.

Smooth projective algebraic varieties are examples of Kähler manifolds. By the Kodaira embedding theorem, Kähler manifolds that have an integral Kähler class can always be embedded into projective spaces.

Kähler metrics are named after German mathematician Erich Kähler, who in 1933 observed that they admit local Kähler potentials, and that for instance the Ricci curvature of the metric can be expressed particularly simply in terms of the potential. This makes the problem of finding Einstein metrics more tractable on Kähler manifolds than on general manifolds.

Definition
Let X be a complex manifold, with complex structure J, that is, at each point $$x \in X$$, J defines an endomorphism of the tangent space $$T_x X$$ corresponding to multiplication by the imaginary unit. A hermitian metric h on X can be decomposed into real and imaginary parts as $$h = g - i\omega$$. Then $$g$$ is a Riemannian metric and $$\omega$$ is an alternating 2-form, that are compatible with the complex structure and each other, in the sense that $$g(Jv, Jw) = g(v,w), \omega(Jv, Jw) = \omega(v,w), \omega(v,w) = g(Jv, w)$$. (The middle condition is equivalent to saying that $$\omega$$ has type (1,1).)


 * Definition: The hermitian metric h is a Kähler metric if the 2-form $$\omega$$ is closed, $$d\omega = 0$$. A Kähler manifold is a complex manifold equipped with a Kähler metric.

Given a Kähler manifold $$(X, J, h)$$, $$(X,g)$$ is a Riemannian manifold, $$(X,\omega)$$ is a symplectic manifold, and $$(X,J)$$ is a complex manifold. Moreover, the condition $$\omega(v,w) = g(v, Jw)$$ means that any two of the structures $$g, \omega, J$$ determines the third one (and hence also h).

The following conditions on the hermitian metric h are equivalent to the Kähler condition $$d\omega = 0$$, and give alternative expressions for the compatibility between the Riemannian and complex structures.
 * The complex structure J is parallel with respect to the Levi-Civita connection of the Riemannian metric g
 * The Levi-Civita connection defined by the Riemannian metric g on the tangent bundle of X is equal to the Chern connection defined by the complex structure and the hermitian metric h

Often the term Kähler manifold refers to a complex manifold that admits a Kähler metric, rather than a complex manifold equipped with a Kähler metric.

The Laplacians on Kähler manifolds
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


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

The Kähler identities imply that


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

It follows that a differential form $$\alpha$$ is harmonic if and only if each of its complex type components $$\alpha^{i,j}$$ are, yielding the Hodge decomposition (see Hodge theory)


 * $$\mathbf{H^r}=\bigoplus_{p+q=r}\mathbf{H}^{p,q}$$

where $$\mathbf{H^r}$$ is the space of r-degree harmonic forms and $$\mathbf{H}^{p,q}$$ is the space of (p,q)-degree harmonic forms on X.

Further, if X is compact then we obtain
 * $$\mathbf{H}^{p,q} \simeq H^{p,q}_{\bar{\partial}}(X)$$

where $$H^{p,q}_{\bar{\partial}}(X)$$ is the Dolbeault cohomology (in turn isomorphic to the sheaf cohomology $$H^p(X,\Omega^q)$$ by the 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.

The Hodge decomposition imposes topological restrictions on which manifolds can admit Kähler metrics. For instance, the Hopf manifold $$(\mathbb{C}^2 \setminus \{0\})/\mathbb{Z}$$ (where $$n \in \mathbb{Z}$$ acts on $$\mathbb{C}^2 \setminus \{0\}$$ by $$x \mapsto \lambda^n x$$ for some non-zero $$\lambda$$) is topologically $$S^1 \times S^3$$, so has $$b_1 = 1$$. On the other hand, by the above any Kähler manifold has $$b_1 = h^{1,0} + h^{0,1} = 2h^{1,0}$$ (and more generally, all odd-degree Betti numbers of a closed Kähler manifold must be even). Thus the Hopf manifold cannot admit any Kähler metrics.

Another consequence of the Hodge decomposition is the $$i \partial \bar \partial$$-lemma: any exact (p+1, q+1)-form $$\alpha$$ on a closed Kähler manifold X can be written as $$i \partial \bar \partial \eta $$ for some (p,q)-form $$\eta$$ (and $$\eta$$ can be taken to be real if $$\alpha$$ is). Griffiths, Morgan, Deligne and Sullivan deduced from this that any closed Kähler manifold must be formal in the sense of rational homotopy theory. In particular, this gives a further topological restriction on which manifolds can admit Kähler metrics.

Calibrations
The standard hermitian metric on $$\mathbb{C}^n$$ has $$\omega = \frac i2 \sum_j dz_j \wedge d\bar{z_j} $$. One can readily compute that $$\frac{\omega^n}{n!}$$ equals the volume form. In the same way, the restriction of $$\frac{\omega^k}{k!}$$ to any k-dimensional complex subspace $$V \subseteq \mathbb{C}^n$$ equals the volume form of V. Wirtinger's inequality states that in fact the restriction of $$\frac{\omega^k}{k!}$$ to any 2k-dimensional oriented real subspace V is less than or equal to the volume form of V, with equality precisely when V is a complex subspace. This means that $$\frac{\omega^k}{k!}$$ is a linear calibration.

A hermitian metric $$ = g - i \omega$$ on a complex manifold X is equivalent on each tangent space to the standard metric on $$\mathbb{C}^n$$. Therefore the value of $$\frac{\omega^k}{k!}$$ at each $$x \in X$$ is a linear calibration on $$T_x X$$. If the metric is Kähler, then $$\frac{\omega^k}{k!}$$ is closed too, so satisfies both conditions for being a calibration on the Riemannian manifold $$(M,g)$$. The calibrated submanifolds are precisely the complex k-dimensional submanifolds of X. Kähler manifolds are the prototypical example of a calibrated geometry.

Local coordinate expression
$$\omega$$ 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 $$ K $$.

Kähler classes
Since the Kähler form $$\omega$$ of a Kähler metric is closed, it represents a de Rham cohomology class. The de Rham cohomology classes on a complex manifold X that can be represented by a Kähler form are called Kähler classes.

If X is closed, so that the Hodge decomposition is defined, then the Kähler classes lie in the $$H^{1,1}$$ part of the decomposition. For a (1,1)-form to be positive (the imaginary part of a hermitian metric) is an open, convex and scale-invariant condition, so the Kähler classes form an open convex cone in $$H^{1,1}(X,\mathbb{R})$$, the Kähler cone.

Note that a Kähler class on a closed manifold is never trivial. For if Kähler form $$\omega$$ is exact, then so is $$\omega^n$$, and its integral would be zero by Stokes theorem. But $$\int_X \omega^n = n! \, Vol(X)$$, which must be positive.

Kähler potentials
For any smooth function $$ \rho : X \to \mathbb{R}$$ on a complex manifold X,
 * $$\frac i2 \partial \bar\partial \rho $$

is a smooth closed (1,1)-form, for $$\partial, \bar \partial$$ the Dolbeault operators. If a Kähler form $$\omega$$ can be written in this form, then $$\rho$$ is called a Kähler potential for $$\omega$$ (and $$\rho$$ is strictly plurisubharmonic).

In fact, as a consequence 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.

On a closed manifold, a Kähler form can never be exact, so can never have a global Kähler potential. On the other and, for two Kähler metrics $$\omega$$ and $$\omega'$$ in the same Kähler class, the difference $$\omega - \omega'$$ is an exact (1,1)-form, so can be written as $$i \partial \bar \partial \phi$$ for a real function $$\phi$$ by the $$i \partial \bar \partial$$-lemma.

Ricci tensor and Kähler manifolds
A Kähler metric is called Kähler–Einstein (or sometimes Einstein–Kähler) if its Ricci tensor is proportional to the metric tensor, $$R = \lambda g$$, for some constant λ (see article on Einstein manifolds). Erich Kähler already observed that the Ricci curvature of a Kähler metric has a simple expression in terms of Kähler potentials.

Further, there is a linear relation between the Ricci curvature of a Kähler metric and the curvature of the induced connection on the canonical bundle. By Chern-Weil theory, the latter determines the first Chern class (with real coefficients) of the complex manifold, which is a topological invariant (see Kähler manifolds in Ricci tensor). In particular, for a complex manifold X to admit a Ricci-flat Kähler metric it is necessary that $$c_1(X) = 0 \in H^2(X;\mathbb{R})$$. In 1977, Shing-Yung Yau proved the Calabi conjecture, which implies in particular that any Kähler class on a closed complex manifold with $$c_1(X) = 0 \in H^2(X;\mathbb{R})$$ contains a unique Ricci-flat Kähler metric (or Calabi-Yau metric).

Riemannian holonomy
The Levi-Civita connection of a Kähler metric on a complex manifold preserves the complex structure, that is, the complex structure is parallel. Therefore Kähler manifolds of complex dimension n are equivalent to Riemannian manifolds with holonomy group contained in the unitary group U(n), the subgroup of the orthogonal group O(2n) that preserves a complex structure. Further, Ricci-flat Kähler metrics (Calabi-Yau manifolds) have holonomy contained in the special unitary group SU(n). The groups U(n) and SU(n) are two of the families in the Berger's classification of Riemannian holonomy groups.

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).