A Kähler structure on a complex manifold 
combines a Riemannian
metric on the underlying real manifold with
the complex structure. Such a structure brings
together geometry and complex analysis, and the main examples come from algebraic
geometry. When 
has 
complex dimensions, then it has 
real dimensions. A Kähler structure is related to the
unitary group 
, which embeds in 
as the orthogonal matrices that preserve the almost
complex structure (multiplication by '
'). In a coordinate chart,
the complex structure of 
defines a multiplication by 
and the metric defines orthogonality for tangent vectors.
On a Kähler manifold, these two notions (and their derivatives) are related.
The following are elements of a Kähler structure, with each condition sufficient for a Kähler structure to exist.
1. A Kähler metric. Near any point 
, there exists holomorphic coordinates 
such that the metric has the form
 |
(1)
|
where 
denotes the vector space tensor product;
that is, it vanishes up to order two at 
. Hence any geometric equation in 
involving only the first derivatives can be defined on a
Kähler manifold. Note that a generic metric
can be written to vanish up to order two, but not necessarily in holomorphic coordinates,
using a Gaussian coordinate system.
2. A Kähler form 
is a real closed nondegenerate
two-form, i.e., a symplectic
form, for which 
for nonzero tangent vectors 
. Moreover, it must also satisfy 
, where 
is the almost complex
structure induced by multiplication by 
. That is,
 |
(2)
|
and
 |
(3)
|
Locally, a Kähler form can be written as 
, where 
is a function called a Kähler
potential. The Kähler form is a real 
-complex form.
3. A Hermitian metric 
where the real part
is a Kähler metric, as in item (1) above,
and where the imaginary part is a Kähler
form, as in item (2).
4. A metric for which the almost complex structure 
is parallel. Since parallel
transport is always an isometry, a Hermitian
metric is well-defined by parallel transport
along paths from a base point. The holonomy group
is contained in the unitary group.
It is easy to see that a complex submanifold of a Kähler manifold inherits its Kähler structure, and so must also be Kähler. The main source of examples are projective algebraic varieties, complex submanifolds of complex projective space which are solutions to algebraic equations.
There are several deep consequences of the Kähler condition. For example, the Kähler identities, the Hodge decomposition of cohomology, and the Lefschetz theorem depends on the Kähler condition for compact manifolds.
