A Kähler form is a closed differential 2-form 
on a complex
manifold 
which is also the negative imaginary part of a
Hermitian metric 
. 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 Riemannian
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 Riemannian
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 real-valued 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 Riemannian metric. A nonsingular
projective algebraic variety admits a Kähler
form representing an integral cohomology class,
meaning that its integral over every integral two-cycle
is an integer. An arbitrary Kähler form on such
an algebraic variety need not have integral
periods. 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
compact Kähler
manifolds of every complex dimension
at least four which do not have the homotopy type
of a nonsingular projective algebraic variety.
In particular, they cannot be deformed to such an algebraic
variety (Voisin 2004).
A Kähler form satisfies Wirtinger's inequality,
 |
(5)
|
where the right-hand side is the volume of the parallelogram formed by the manifold tangent vectors 
and 
. Corresponding inequalities
hold for the exterior powers of 
. For linearly independent 
and 
, equality holds iff
they 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 
.
