A closed two-form 
on a complex
manifold 
which is also the negative imaginary part of a
Hermitian metric 
is called a Kähler form. In this case, 
is called a Kähler
manifold and 
,
the real part of the Hermitian
metric, is called a Kähler metric. The
Kähler form combines the metric and the complex
structure, indeed
 |
(1)
|
where 
is the almost complex structure induced
by multiplication by 
.
Since the Kähler form comes from a Hermitian
metric, it is preserved by 
, i.e., since 
. The equation 
implies that the metric and the complex structure are
related. It gives 
a Kähler structure, and has many implications.
On 
, the Kähler form can be written
as
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
.
In general, the Kähler form can be written in coordinates
 |
(4)
|
where 
is a Hermitian metric, the real
part of which is the Kähler metric. Locally,
a Kähler form can be written as 
, where 
is a function called a Kähler
potential. The Kähler form is a real 
-complex form.
Since the Kähler form 
is closed, it represents a cohomology
class in de Rham cohomology. On a compact
manifold, it cannot be exact because 
is the volume form determined by the metric. In
the special case of a projective algebraic
variety, the Kähler form represents an integral
cohomology class. That is, it integrates to an integer on any one-dimensional
submanifold, i.e., an algebraic curve. The Kodaira embedding theorem says that if the
Kähler form represents an integral cohomology
class on a compact manifold, then it must be a projective
algebraic variety. There exist Kähler forms which are not projective algebraic,
but it is an open question whether or not any Kähler
manifold can be deformed to a projective
algebraic variety (in the compact case).
A Kähler form satisfies Wirtinger's inequality,
 |
(5)
|
where the right-hand side is the volume of the parallelogram formed by the tangent vectors 
and 
. Corresponding inequalities hold for
the exterior powers of 
. Equality holds iff 
and 
form a complex subspace. Therefore, 
is a calibration form,
and the complex submanifolds of a Kähler manifold are calibrated submanifolds. In particular, the complex submanifolds
are locally volume minimizing in a Kähler manifold. For example, the graph of
a holomorphic function is a locally area-minimizing surface in 
.
