Kähler form - Imported Knowlpedia Content

Definition

Let $(M,J)$ be a complex manifold. A Kähler form is a real differential $2$ -form $\omega$ such that

d\omega=0,\qquad \omega(JX,JY)=\omega(X,Y),\qquad \omega(X,JX)>0

for every nonzero real tangent vector $X$ . The second condition says that $\omega$ has complex type $(1,1)$ , and the third is positivity. These conditions make $\omega$ nondegenerate, so it is also a symplectic form. The formula

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

defines a $J$ -invariant Riemannian metric, called the associated Kähler metric.

Equivalent metric formulation

Conversely, the fundamental form $\omega(X,Y)=g(JX,Y)$ of a Hermitian metric is a Kähler form exactly when it is closed. Thus specifying a Kähler form on $(M,J)$ is equivalent to specifying a Kähler metric, and it makes $(M,J,g)$ a Kähler manifold Voisin, §3.1.

Local potentials

Every Kähler form is locally expressible as

\omega=i\,\partial\bar\partial\varphi

for a real strictly plurisubharmonic function $\varphi$ , with a constant factor changed under other $d^c$ conventions. Adding the real part of a holomorphic function to $\varphi$ does not change $\omega$ . This potential description is local; a global potential need not exist.

Examples and conventions

The standard form $\frac{i}{2}\sum_j dz^j\wedge d\bar z^j$ on $\mathbb C^n$ is Kähler, as is the Fubini–Study form on complex projective space. Some authors use $\omega(X,Y)=g(X,JY)$ , which reverses the sign relative to the convention here and therefore changes the corresponding positivity inequality.

References

Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge University Press, 2002. DOI record. Relevant: §3.1, Kähler metrics and forms.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: Chapter 3, Kähler forms and local potentials.