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Kähler Form


A closed two-form omega on a complex manifold M which is also the negative imaginary part of a Hermitian metric h=g-iomega is called a Kähler form. In this case, M is called a Kähler manifold and g, the real part of the Hermitian metric, is called a Kähler metric. The Kähler form combines the metric and the complex structure, indeed

 g(X,Y)=omega(X,JY),
(1)

where J is the almost complex structure induced by multiplication by i. Since the Kähler form comes from a Hermitian metric, it is preserved by J, i.e., since h(X,Y)=h(JX,JY). The equation domega=0 implies that the metric and the complex structure are related. It gives M a Kähler structure, and has many implications.

On C^2, the Kähler form can be written as

omega=-1/2i(dz_1 ^ dz_1^_+dz_2 ^ dz_2^_)
(2)
=dx_1 ^ dy_1+dx_2 ^ dy_2,
(3)

where z_n=x_n+iy_n. In general, the Kähler form can be written in coordinates

 omega=sumg_(ik^_)dz_i ^ dz^__k,
(4)

where g_(ik^_) is a Hermitian metric, the real part of which is the Kähler metric. Locally, a Kähler form can be written as partialpartial^_f, where f is a function called a Kähler potential. The Kähler form is a real (1,1)-complex form.

Since the Kähler form omega is closed, it represents a cohomology class in de Rham cohomology. On a compact manifold, it cannot be exact because omega^n/n!!=0 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,

 |omega(X,Y)|<=|X ^ Y|,
(5)

where the right-hand side is the volume of the parallelogram formed by the tangent vectors X and Y. Corresponding inequalities hold for the exterior powers of omega. Equality holds iff X and Y form a complex subspace. Therefore, omega 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 C^2=R^4.


See also

Calabi-Yau Space, Calibration Form, Complex Manifold, Complex Projective Space, Dolbeault Cohomology, Kähler Identities, Kähler Manifold, Kähler Metric, Kähler Potential, Kähler Structure, Kodaira Embedding Theorem, Projective Algebraic Variety, Symplectic Form, Wirtinger's Inequality

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Kähler Form." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/KaehlerForm.html

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