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)