- Kähler manifold
In
mathematics , a Kähler manifold is amanifold with unitary structure (a "U"("n")-structure) satisfying anintegrability condition .In particular, it is acomplex manifold , aRiemannian manifold , and asymplectic 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 analmost Kähler manifold ; if the complex structure is integrable (but the Sp-structure need not be), the notion is aHermitian manifold .Kähler manifolds are named for the mathematician
Erich Kähler and are important inalgebraic geometry : they are a differential geometric generalization of complex algebraic varietiesDefinition
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 thetangent bundle TM satisfying a condition that has several equivalent characterizations (the most geometric being thatparallel 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 :h = sum h_{iar j}; dz^i otimes d ar z^jis 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: 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}^{jfor 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 ofEinstein 's considerations about thecosmological constant . See the article onEinstein manifold s 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 acompact Kähler manifold.
#Every Riemannian metric on aRiemann 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, theFubini-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 acomplex submanifold of a Kähler manifold is Kähler. In particular, anyStein manifold (embedded in C"n") oralgebraic 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 theBergman metric which has constant holomorphic sectional curvature.
#EveryK3 surface is Kähler (by a theorem of Y.-T. Siu).An important subclass of Kähler manifolds are
Calabi–Yau manifold s.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.
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