# Kähler form

The fundamental form of a Kähler metric on a complex manifold. A Kähler form is a harmonic real differential form of type $ ( 1, 1) $.
A differential form $ \omega $
on a complex manifold $ M $
is the Kähler form of a Kähler metric if and only if every point $ x \in M $
has a neighbourhood $ U $
in which

$$ \omega = \ i \partial \overline \partial \; p = i \sum \frac{\partial ^ {2} p }{\partial z _ \alpha \partial \overline{z}\; _ \beta } dz _ \alpha \wedge d \overline{z}\; _ \beta , $$

where $ p $ is a strictly plurisubharmonic function in $ U $ and $ z _ {1} \dots z _ {n} $ are complex local coordinates.

A Kähler form is called a Hodge form if it corresponds to a Hodge metric, i.e. if it has integral periods or, equivalently, defines an integral cohomology class.

#### References

[1] | R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) |

#### Comments

For fundamental form of a Kähler metric see Kähler metric.

#### References

[a1] | A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958) |

**How to Cite This Entry:**

Kähler form.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=K%C3%A4hler_form&oldid=47539