Kähler Form

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 .