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.

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[edit] Definition

A Kähler metric on a complex manifold M is a hermitian metric on the tangent bundle TM 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_{i\bar j}\; 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_{i\bar j}\; 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_{i\bar{j}} = \frac{\partial^2 K}{\partial z^i \partial \bar{z}^{j}}

for some function K, called the Kähler potential.

A Kähler manifold, the associated Kähler form and metric are called Kähler-Einstein (or sometimes Einstein-Kähler) iff its Ricci tensor is proportional to the metric tensor, Ric = λ g, for some constant λ. This name is a reminder of Einstein's considerations about the cosmological constant. See the article on Einstein manifolds for more details.

[edit] Examples

  1. Complex Euclidean space Cn with the standard Hermitian metric is a Kähler manifold.
  2. 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.
  3. Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 (real) dimensions.
  4. 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.
  5. 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.
  6. 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.

[edit] See also

[edit] 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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