Kähler manifold - Knowlpedia

Let $(M,J,g)$ be a Hermitian manifold, and let

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

be its fundamental $2$ -form. The manifold is Kähler if

d\omega=0.

Thus a Kähler manifold is a complex manifold with a $J$ -invariant real Riemannian metric whose associated real $2$ -form is closed. The metric $g$ determines a sesquilinear Hermitian metric on the complex tangent bundle, but $g$ itself remains a real bilinear form. The sign used for $\omega$ does not affect the closedness condition.

Equivalent characterizations

For a Hermitian manifold, the Kähler condition is equivalent to $\nabla J=0$ , where $\nabla$ is the Levi–Civita connection of $g$ . Equivalently, parallel transport preserves both $g$ and $J$ , so the holonomy acts through $U(n)$ .

Structure and consequences

The closed, nondegenerate form $\omega$ makes $M$ a symplectic manifold. In complex dimension $n$ , the form $\omega^n/n!$ is the Riemannian volume form for the compatible orientation. Kähler geometry therefore ties together complex, Riemannian, and symplectic structures without making those structures identical.

Morphisms

There is no single convention for a “Kähler morphism,” so the preserved data must be named. A holomorphic map preserves the complex structures but need not preserve the metrics or Kähler forms. A holomorphic isometric immersion, also called a Kähler immersion, additionally satisfies $f^*g_N=g_M$ , equivalently $f^*\omega_N=\omega_M$ . If such a map is a diffeomorphism, it is simultaneously a biholomorphism, Riemannian isometry, and symplectomorphism.

Examples and non-examples

Complex Euclidean space, complex tori with flat Hermitian metrics, and complex projective space with the Fubini–Study metric are Kähler. In complex dimension greater than one, conformally replacing a Kähler metric by $g'=e^f g$ for a nonconstant real function gives $\omega'=e^f\omega$ and

d\omega'=e^f\,df\wedge\omega,

which is nonzero where $df\neq0$ ; the new metric is Hermitian but not Kähler. In complex dimension one, every Hermitian metric is Kähler because every real $3$ -form vanishes.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: §3.1, Definition 3.1.6 for Kähler closedness, and Appendix 4.A for the Levi–Civita and holonomy characterization.