nLab
Kähler manifold

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Complex geometry

Contents

Idea

A Kähler manifold is a smooth manifold compatibly equipped with

  1. complex structure;

  2. Riemannian structure;

  3. symplectic structure.

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

complex structure+ Riemannian structure+ symplectic structure
complex structureHermitian structureKähler structure

Where a Riemannian manifold is a real smooth manifold equipped with a nondegenerate smooth symmetric 2-form gg (the Riemannian metric), an almost Kähler manifold is a complex holomorphic manifold equipped with a nondegenerate hermitian 2-form hh (the Kähler 22-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+iωh = g+i\omega; here gg is a Riemannian metric and ω\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 n/L\mathbb{C}^n/L where LL is a lattice in n\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)(X, J) is an almost complex manifold, then there is a notion of smooth complex-valued differential forms of type (p,q)(p,q). A complex valued 22-form ω\omega is of type (1,1)(1,1) precisely if it satisfies

ω(Jv,Jw)=ω(v,w) \omega(J v,J w) = \omega(v,w)

for all smooth vector fields v,wv,w on XX. Here ω\omega is a real 22-form of type (1,1)(1,1), if ω¯=ω\overline \omega = \omega. Setting

g(v,w)=ω(v,Jw), g(v,w) = \omega(v, J w),

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

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

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

Hence given a complex manifold XX, together with a closed real 22-form ω\omega, the only additional condition required to ensure that it defines a Kähler metric is that it be a positive (1,1)(1,1)-form.

Relation to Spin-structures

Proposition

A spin structure on a compact Hermitian manifold (Kähler manifold) XX of complex dimension nn exists precisely if, equivalently

In this case one has:

Proposition

There is a natural isomorphism

S XΩ X 0,Ω n,0 X S_X \simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{n,0}}_X

of the sheaf of sections of the spinor bundle S XS_X on XX 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 Σ\Sigma of dimension dim (Σ)=ndim_{\mathbb{C}}(\Sigma) = n the Hodge star operator acts on the Dolbeault complex as

:Ω p,q(X)Ω nq,np(X). \star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X) \,.

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

Hodge structure

The Hodge theorem asserts that for a compact Kähler manifold, the canonical (p,q)(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.

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
Kähler manifoldU(k)2k2kKähler forms
Calabi-Yau manifoldSU(k)2k2k
hyper-Kähler manifoldSp(k)4k4k
G2 manifoldG277associative 3-form
Spin(7) manifoldSpin(7)8Cayley 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)

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 June 16, 2014 03:25:11 by Urs Schreiber (88.128.80.80)