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 Riemannian metric defines orthogonality
for manifold 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 exist holomorphic coordinates 
such that the Riemannian
metric has the form
 |
(1)
|
where 
denotes the vector space tensor product;
that is, the difference between 
and its Euclidean leading term vanishes to order two at 
. Hence any geometric equation
in 
involving only the first derivatives can be defined
on a Kähler manifold. A generic Riemannian
metric has the same Euclidean leading term with an error vanishing to order two
in normal coordinates, but these coordinates
need not be holomorphic.
2. A Kähler form 
is a real closed nondegenerate
differential 2-form, i.e., a symplectic
form, for which 
for nonzero manifold
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 real-valued 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 negative of the imaginary part is
a Kähler form, as in item (2).
4. A Hermitian metric compatible with the almost complex structure 
for which 
is parallel with respect to the Levi-Civita
connection. 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 consists of projective smooth 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 hard Lefschetz theorem depend on the Kähler condition for compact manifolds.
