Kähler manifold

In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. Kähler manifolds can thus be thought of as Riemannian manifolds and symplectic manifolds in a natural way.

Kähler manifolds are named for the mathematician Erich Kähler and are important in algebraic geometry.

Definition

A Kähler metric on a complex manifold M is a hermitian metric on the complexified tangent bundle TM⊗C satisfying a condition that has several equivalent characterizations (the most geometric being that parallel transport gives rise to complex-linear mappings on the tangent spaces). In terms of local coordinates it is specified in this way: if

is the hermitian metric, then the associated Kähler form (defined up to a factor of i/2) by

is closed: that is, dω = 0. If M carries such a metric it is called a Kähler manifold.

Examples

1. Complex Euclidean space Cⁿ with the standard hermitian metric is a Kähler manifold.
2. A complex torus, given by Cⁿ/Λ for some lattice Λ, forms a compact Kähler manifold with the natural metric.
3. Every Riemann surface is a Kähler manifold, since the condition for ω to be closed is trivial in 2 (real) dimensions.
4. Complex projective space CPⁿ has a natural Kähler metric called the Fubini-Study metric. It is essentially determined by the condition that it be invariant under the action of the unitary group (of dimension one larger, acting on the complex vector space giving rise to the projective space).
5. Any complex submanifold of a Kähler manifold is Kähler. In particular, any complex manifold that can be embedded in Cⁿ or CPⁿ is Kähler.
6. The restriction properties of the Fubini-Study metric mean that non-singular projective complex algebraic varieties carry Kähler metrics. This is fundamental to their analytic theory.

An important subclass of Kähler manifolds are Calabi-Yau manifolds.