Kähler manifold

In mathematics, a Kähler manifold is a manifold with unitary structure (a "U"("n")-structure) satisfying an integrability condition.In particular, it is a complex manifold, a Riemannian manifold, and a symplectic manifold, with these three structures all mutually compatible.

This threefold structure corresponds to the presentation of the unitary group as an intersection::$U(n)\; =\; O(2n)\; cap\; GL(n,mathbf\{C\})\; cap\; Sp(2n).$

Without any integrability conditions, the analogous notion is an almost Hermitian manifold. If the Sp-structure is integrable (but the complex structure need not be), the notion is an almost Kähler manifold; if the complex structure is integrable (but the Sp-structure need not be), the notion is a Hermitian manifold.

Kähler manifolds are named for the mathematician Erich Kähler and are important in algebraic geometry: they are a differential geometric generalization of complex algebraic varieties

Definition

A manifold with a Hermitian metric is an almost Hermitian manifold; a Kähler manifold is a manifold with a Hermitian metric that satisfies an integrability condition, which has several equivalent formulations.

Kähler manifolds can be characterized in many ways: they are often defined as a complex manifold with an additional structure (or a symplectic manifold with an additional structure, or a Riemannian manifold with an additional structure).

One can summarize the connection between the three structures via $h=g\; +\; iomega$, where "h" is the Hermitian form, "g" is the Riemannian metric, "i" is the almost complex structure, and $omega$ is the almost symplectic structure.

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 induced by the metric 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.

The metric on a Kähler manifold locally satisfies: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, $R\; =\; lambda\; g$, 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.

Examples

#Complex Euclidean space C^"n" with the standard Hermitian metric is a Kähler manifold.
#A torus C^"n"/Λ (Λ a full lattice) 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 Bergman metric which has constant holomorphic sectional curvature.
#Every K3 surface is Kähler (by a theorem of Y.-T. Siu).

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

ee also

*Almost complex manifold
*Hyper-Kähler manifold
*Kähler–Einstein metric
*Quaternion-Kähler manifold
*Complex Poisson manifold
*Calabi-Yau manifold

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.
*Andrei Moroianu, "Lectures on Kähler Geometry" (2007), London Mathematical Society Student Texts 69, Cambridge ISBN 978-0-521-68897-0.