A complex manifold for which the exterior derivative of the fundamental form associated with the given Hermitian
metric vanishes, so . In other words, it is a complex manifold with a Kähler structure. It has a Kähler
form, so it is also a symplectic manifold.
It has a Kähler metric, so it is also a Riemannian manifold.

The simplest example of a Kähler manifold is a Riemann surface, which is a complex manifold of dimension
1. In this case, the imaginary part of any Hermitian
metric must be a closed form since all 2-forms
are closed on a two real dimensional manifold.

## See also

Calibrated Manifold,

Complex Manifold,

Complex Projective Space,

Hyper-Kähler Manifold,

Kähler
Form,

Kähler Identities,

Kähler
Metric,

Kähler Potential,

Kähler
Structure,

Projective Algebraic Variety,

Quaternion Kähler Manifold Riemannian Metric,

Symplectic
Manifold
*Portions of this entry contributed by Todd
Rowland*

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## Cite this as:

Rowland, Todd and Weisstein, Eric W. "Kähler Manifold." From *MathWorld*--A
Wolfram Web Resource. https://mathworld.wolfram.com/KaehlerManifold.html

## Subject classifications