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
defines a 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
(notice the exchange of the role of and ) See e.g. (BiquerdHöring 08, p. 79).
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.
Textbook accounts include
Lecture notes include
Discussion of spin structures in Kähler manifolds is for instance in
Discussion of Hodge theory on Kähler manifolds is in