# 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)