A Kähler manifold is a smooth manifold compatibly equipped with
If the symplectic structure is not compatibly present, it is just a Hermitian manifold.
|complex structure||+ Riemannian structure||+ symplectic structure|
|complex structure||Hermitian structure||Kähler structure|
Where a Riemannian manifold is a real smooth manifold equipped with a nondegenerate smooth symmetric 2-form (the Riemannian metric), an almost Kähler manifold is a complex holomorphic manifold equipped with a nondegenerate hermitian 2-form (the Kähler -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.
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 where is a lattice in , K3-surfaces, compact Calabi-Yau manifolds, quadrics, products of projective spaces and so on.
for all smooth vector fields on . Here is a real -form of type , if . Setting
definesa smooth symmetric rank tensor field. This is a Riemannian metric precisely if it is fiberwise a positive definite bilinear form. If it is hence a Riemannian metric, then is called positive definite, too.
Hence given a complex manifold , together with a closed real -form , the only additional condition required to ensure that it defines a Kähler metric is that it be a positive -form.
In this case one has:
There is a natural isomorphism
The Hodge theorem asserts that for a compact Kähler manifold, the canonical -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.
|G-structure||special holonomy||dimension||preserved differential form|
|Kähler manifold||U(k)||Kähler forms|
|G2 manifold||G2||associative 3-form|
|Spin(7) manifold||Spin(7)||8||Cayley form|
Kähler manifolds were first introduced and studied by P. A. Shirokov (cf. a historical article) and later independently by Kähler.
Lecture notes include
Discussion of spin structures in Kähler manifolds is for instance in