nLab
Kähler manifold

Idea

Kähler manifolds are analogues in complex geometry of Riemannian manifolds in real differential geometry.

Where a Riemannian manifold is a real smooth manifold equipped with a nondegenerate smooth symmetric 2-form $g$ (the Riemannian metric), an almost Kähler manifold is a complex holomorphic manifold equipped with a nondegenerate hermitian 2-form $h$ (the Kähler $2$-form). The real cotangent bundle is replaced with the complex cotangent bundle, and symmetry is replaced with hermitian symmetry. An almost Kähler manifold is a Kähler manifold if it satisfies an additional integrability condition.

The Kähler 2-form can be decomposed as $h=g+i\omega$; here $g$ is a Riemannian metric and $\omega$ a symplectic form.

Examples

There is a unique up to a scalar hermitian metric on a complex projective space (which can be normalized), the Fubini–Study metric. All analytic subvarieties of a complex projective space are in fact algebraic subvarieties and they inherit the Kähler structure from the projective space. Examples include complex tori ${ℂ}^n/L$ where $L$ is a lattice in ${ℂ}^n$, K3-surfaces, compact Calabi-Yau manifolds, quadrics, products of projective spaces and so on.

Properties

Hodge structure

The Hodge theorem asserts that for a compact Kähler manifold, the canonical $(p,q)$-grading of its differential forms descends to its de Rham cohomology/ordinary cohomology. The resulting structure is called a Hodge structure, and is indeed the archetypical example of such.

Related concepts

• Kähler manifold, hyper-Kähler manifold, quaternionic Kähler manifold

• Kähler-Einstein manifold

classification of special holonomy manifolds by Berger's theorem:

G-structure	special holonomy	dimension	preserved differential form
Kähler manifold	U(k)	$2k$	Kähler forms
Calabi-Yau manifold	SU(k)	$2k$
hyper-Kähler manifold	Sp(k)	$4k$
G2 manifold	G2	$7$	associative 3-form
Spin(7) manifold	Spin(7)	8	Cayley form

• symplectic manifold

• Kähler polarization

References

Kähler manifolds were first introduced and studied by P. A. Shirokov (cf. a historical article) and later independently by Kähler.

Lecture notes include

• Andrei Moroianu, Lectures on Kähler Geometry (pdf)