Kähler metric - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold with complex structure $J$ . A Kähler metric is a Hermitian Riemannian metric $g$ whose associated real fundamental $2$ -form

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

is closed:

d\omega=0.

Equivalently, $\omega$ is a Kähler form, and $(X,J,g)$ is a Kähler manifold with specified metric data. The positivity and $J$ -invariance belong to the Hermitian hypothesis, while closedness is the additional Kähler condition. The metric $g$ is real bilinear; the corresponding Hermitian form on the complex tangent bundle is sesquilinear.

Equivalent characterizations

For a Hermitian metric, the Kähler condition is equivalent to $\nabla J=0$ for the Levi–Civita connection. It is also equivalent to the Levi–Civita and Chern connections on the holomorphic tangent bundle agreeing. In holomorphic coordinates, the Hermitian coefficient matrix $(g_{j\bar k})$ satisfies

\frac{\partial g_{j\bar k}}{\partial z^\ell} = \frac{\partial g_{\ell\bar k}}{\partial z^j}.

The closed-form and local-coordinate conditions are given in Demailly, Chapter VI, §4, Definition 4.1.

Structure and consequences

The fundamental form of a Kähler metric is nondegenerate, so it also makes $X$ a symplectic manifold. The Levi–Civita connection preserves $g$ , $J$ , and $\omega$ , and its holonomy is contained in $U(n)$ . Locally, the metric coefficients are complex Hessians of real Kähler potentials. These compatibilities are special: an arbitrary Hermitian metric need not have closed fundamental form.

Examples and non-examples

The Euclidean metric on $\mathbb C^n$ , flat Hermitian metrics on complex tori, and the Fubini–Study metric on complex projective space are Kähler. If $n>1$ , multiplying a Kähler metric by $e^f$ for a nonconstant real function $f$ usually gives only a Hermitian metric, because its fundamental form is $e^f\omega$ and

d(e^f\omega)=e^f\,df\wedge\omega

need not vanish.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: §3.1, Kähler metrics and their fundamental forms.
Jean-Pierre Demailly, Complex Analytic and Differential Geometry, 2012. Author-hosted text. Relevant: Chapter VI, §4, especially Definition 4.1 and Theorem 4.8.