# nLab Kähler manifold

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

complex geometry

# Contents

## Idea

A Kähler manifold is a smooth manifold compatibly equipped with

If the symplectic structure is not compatibly present, it is just a Hermitian manifold.

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.

## Definition

### In terms of $G$-Structure

A Kähler manifold is a first-order integrable almost Hermitian structure, hence a first order integrable G-structure for $G = U(n) \hookrightarrow GL(2n,\mathbb{R})$ the unitary group (e.g. Verbitsky 09).

By the fact (see at unitary group – relation to orthogonal, symplectic and general linear group) that $U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})$ this means that a Kähler manifold structure is precisely a joint orthogonal structure/Riemannian manifold structure, symplectic manifold structure and complex manifold structure.

## 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 $\mathbb{C}^n/L$ where $L$ is a lattice in $\mathbb{C}^n$, K3-surfaces, compact Calabi-Yau manifolds, quadrics, products of projective spaces and so on.

## Properties

### Relation to (almost) complex manifold

The following based on this MO comment by Spiro Karigiannis

When $(X, J)$ is an almost complex manifold, then there is a notion of smooth complex-valued differential forms of type $(p,q)$. A complex valued $2$-form $\omega$ is of type $(1,1)$ precisely if it satisfies

$\omega(J v,J w) = \omega(v,w)$

for all smooth vector fields $v,w$ on $X$. Here $\omega$ is a real $2$-form of type $(1,1)$, if $\overline \omega = \omega$. Setting

$g(v,w) = \omega(v, J w),$

defines a smooth symmetric rank $(2,0)$ tensor field. This is a Riemannian metric precisely if it is fiberwise a positive definite bilinear form. If it $g(-,-) = \omega(-,J -)$ is hence a Riemannian metric, then $\omega(-,-)$ is called positive definite, too.

The triple of data $(J, \omega, g)$, where $J$ is an almost complex structure, $\omega$ is a real positive $(1,1)$-differential form, and $g$ is the associated Riemannian metric this way define an almost Hermitian manifold.

Now the condition for $X$ to be a Kähler is that $X$ be a complex manifold ($J$ is integrable) and that $d\omega = 0$. Equivalently that for the Levi-Civita connection $\nabla$ of $G$ we have $\nabla \omega = 0$ or $\nabla J = 0$.

Hence given a complex manifold $X$, together with a closed real $2$-form $\omega$, the only additional condition required to ensure that it defines a Kähler metric is that it be a positive $(1,1)$-form.

### Relation to symplectic manifolds

Lifting a symplectic manifold structure to a Kähler manifold structure is also called choosing a Kähler polarization.

### Relation to Spin-structures

###### Proposition

A spin structure on a compact Hermitian manifold (Kähler manifold) $X$ of complex dimension $n$ exists precisely if, equivalently

In this case one has:

###### Proposition

There is a natural isomorphism

$S_X \simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{n,0}}_X$

of the sheaf of sections of the spinor bundle $S_X$ on $X$ with the tensor product of the Dolbeault complex with the corresponding Theta characteristic;

Moreover, the corresponding Dirac operator is the Dolbeault-Dirac operator $\overline{\partial} + \overline{\partial}^\ast$.

This is due to (Hitchin 74). A textbook account is for instance in (Friedrich 74, around p. 79 and p. 82).

### Hodge star operator

On a Kähler manifold $\Sigma$ of dimension $dim_{\mathbb{C}}(\Sigma) = n$ the Hodge star operator acts on the Dolbeault complex as

$\star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X) \,.$

(notice the exchange of the role of $p$ and $q$) See e.g. (BiquerdHöring 08, p. 79).

### 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.

### As $\mathbb{O}$-Riemannian manifolds

normed division algebra$\mathbb{A}$Riemannian $\mathbb{A}$-manifoldsSpecial Riemannian $\mathbb{A}$-manifolds
real numbers$\mathbb{R}$Riemannian manifoldoriented Riemannian manifold
complex numbers$\mathbb{C}$Kähler manifoldCalabi-Yau manifold
quaternions$\mathbb{H}$quaternion-Kähler manifoldhyperkähler manifold
octonions$\mathbb{O}$Spin(7)-manifoldG2-manifold

(Leung 02)

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
$\mathbb{C}$Kähler manifoldU(k)$2k$Kähler forms $\omega_2$
Calabi-Yau manifoldSU(k)$2k$
$\mathbb{H}$quaternionic Kähler manifoldSp(k)Sp(1)$4k$$\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$
hyper-Kähler manifoldSp(k)$4k$$\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2$ ($a^2 + b^2 + c^2 = 1$)
$\mathbb{O}$Spin(7) manifoldSpin(7)8Cayley form
G2 manifoldG2$7$associative 3-form

## References

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

Textbook accounts include

Lecture notes include

• Andrei Moroianu, Lectures on Kähler Geometry ([pdf] (http://www.math.polytechnique.fr/~moroianu/tex/kg.pdf))

Discussion in terms of first-order integrable G-structure include

• Misha Verbitsky, Kähler manifolds, lecture notes 2009 (pdf)

Discussion of spin structures in Kähler manifolds is for instance in

• Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate studies in mathematics 25, AMS (1997)

Discussion of Hodge theory on Kähler manifolds is in

• O. Biquard, A. Höring, Kähler geometry and Hodge theory, 2008 (pdf)

Revised on September 6, 2016 09:53:23 by Urs Schreiber (195.37.209.183)