Kähler manifold

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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.

Contents

1 Definition
2 Examples
3 See also
4 References

Definition

A Kähler metric on a complex manifold M is a hermitian metric on the complexified tangent bundle [ TM \otimes \mathbb 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_\; dz^i \otimes d \bar z^j]

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

[\omega = \sum h_\; dz^i \wedge d \bar z^j ]

is closed: 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_} = \frac^}]

for some function K, called the Kähler potential.

Examples

Complex Euclidean space Cⁿ with the standard Hermitian metric is a Kähler manifold.
A torus Cⁿ/Λ (Λ a full lattice) inherits a flat metric from the Euclidean metric on Cⁿ, 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 CPⁿ admits a homogeneous Kähler metric, the Fubini-Study metric. An Hermitian form in (the vector space) Cⁿ⁺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 CPⁿ, 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 Cⁿ) or algebraic variety (embedded in CPⁿ) is of Kähler type. This is fundamental to their analytic theory.
The unit complex ball Bⁿ admits a Kähler metric called the Bergman metric which has constant holomorphic sectional curvature.

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

References

André Weil, Introduction à l'étude des variétés kählériennes (1958)
Alan Huckleberry and Tilman Wurzbacher, eds. Infinite Dimensional Kähler Manifolds (2001), Birkhauser Verlag, Basel ISBN 3-7643-6602-8.

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Kähler manifold

Definition

Examples

See also

References