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 otimes mathbf 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 :h = sum h_{iar j}; dz^i otimes d ar z^j is the hermitian metric, then the associated Kähler form (defined up to a factor of i /2) by :omega = sum h_{iar j}; dz^i wedge d ar z^j is closed form: that is, dω = 0. If M carries such a metric it is called a Kähler manifold.

The metric on a Kähler manifold locally satisfies :g_{iar{j}} = frac{partial^2 K}{partial z^i partial ar{z}^{j}} for some function K , called the Kähler potential.

=Examples=

#Complex Euclidean space C n with the standard Hermitian metric is a Kähler manifold. #A torus C n /Λ (Λ a lattice (group)) inherits a flat metric from the Euclidean metric on C n , and is therefore a compact Kähler manifold. #Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 (real) dimensions. #Complex projective space CP n admits a homogeneous Kähler metric, the Fubini-Study metric. An Hermitian form in (the vector space) C n+1 defines a unitary subgroup U(n+1) in GL(n+1,C) ; a Fubini-Study metric is determined up to homothety (overall scaling) by invariance under such a U(n+1) action. By elementary linear algebra, any two Fubini-Study metrics are isometric under a projective automorphism of CP n , so it is common to speak of the Fubini-Study metric. #The induced metric on a complex submanifold of a Kähler manifold is Kähler. In particular, any Stein manifold (embedded in C n ) or algebraic variety (embedded in CP n ) is of Kähler type. This is fundamental to their analytic theory. #The unit complex ball B n admits a Kähler metric called the Bergmann metric which has constant holomorphic sectional curvature.

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

=See also=

*almost complex manifold *complex manifold *Hermitian manifold *hyper-Kähler manifold *quaternion-Kähler manifold

=References=