Kähler manifold
Encyclopedia
In mathematics
, a Kähler manifold is a manifold
with unitary
structure (a U(n)-structure
) satisfying an integrability condition.
In particular, it is a Riemannian manifold
, a complex 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:
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 after the mathematician Erich Kähler
and are important in algebraic geometry
: they are a differential geometric generalization of complex algebraic varieties
.
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
, where h is the Hermitian form, g is the Riemannian metric, i is the almost complex structure
, and
is the almost symplectic structure
.
A Kähler metric on a complex manifold M is a hermitian metric on the tangent bundle
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,
, for some constant λ. This name is a reminder of Einstein's considerations about the cosmological constant
. See the article on Einstein manifold
s for more details.
An important subclass of Kähler manifolds are Calabi–Yau manifolds.
s vanish on a Kähler manifold. Manifolds with such vanishing are formal
: their real homotopy type follows ("formally") from their real cohomology ring.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Kähler manifold is a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
with unitary
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
structure (a U(n)-structure
G-structure
In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....
) satisfying an integrability condition.
In particular, it is a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
, a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
, and a symplectic manifold
Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
, with these three structures all mutually compatible.
This threefold structure corresponds to the presentation of the unitary group as an intersection:
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
Hermitian manifold
In mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. Specifically, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each tangent space...
.
Kähler manifolds are named after the mathematician Erich Kähler
Erich Kähler
was a German mathematician with wide-ranging geometrical interests.Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig. He held professorial positions in Königsberg, Leipzig, Berlin and Hamburg...
and are important in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
: they are a differential geometric generalization of complex algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
.
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
, where h is the Hermitian form, g is the Riemannian metric, i is the almost complex structureAlmost complex manifold
In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost...
, and
is the almost symplectic structureAlmost symplectic manifold
In differential geometry, an almost symplectic structure on a differentiable manifold M is a two-form ω on M which is everywhere non-singular. If, in addition, ω is closed, then it is a symplectic form....
.
A Kähler metric on a complex manifold M is a hermitian metric on the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
satisfying a condition that has several equivalent characterizations (the most geometric being that parallel transportParallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
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
Closed form
-Maths:* Closed-form expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0-Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...
: 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,
, for some constant λ. This name is a reminder of Einstein's considerations about the cosmological constantCosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
. See the article on Einstein manifold
Einstein manifold
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric...
s for more details.
Examples
- Complex Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Cn with the standard Hermitian metric is a Kähler manifold. - A torus Cn/Λ (Λ a full latticeLattice (group)In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
) inherits a flat metric from the Euclidean metric on Cn, and is therefore a compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Kähler manifold. - Every Riemannian metric on a Riemann surfaceRiemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
is Kähler, since the condition for ω to be closed is trivial in 2 (real) dimensions. - Complex projective spaceComplex projective spaceIn mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
CPn admits a homogeneous Kähler metric, the Fubini-Study metricFubini-Study metricIn mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study....
. 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 manifoldStein manifoldIn mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...
(embedded in Cn) or projective algebraic varietyAlgebraic varietyIn mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
(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 metricBergman metricIn differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold. It is so called because it is derived from the Bergman kernel, both of which are named for Stefan Bergman.-Definition:...
which has constant holomorphic sectional curvature. - Every K3 surfaceK3 surfaceIn mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
is Kähler (by a theorem of Y.-T. Siu).
An important subclass of Kähler manifolds are Calabi–Yau manifolds.
Properties
showed that all Massey productMassey product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product.-Massey triple product:...
s vanish on a Kähler manifold. Manifolds with such vanishing are formal
Formal manifold
In geometry and topology, a formal manifold can mean one of a number of related concepts:* In the sense of Dennis Sullivan, a formal manifold is one whose real homotopy type is a formal consequence of its real cohomology ring; algebro-topologically this means that all Massey products vanish.* A...
: their real homotopy type follows ("formally") from their real cohomology ring.
See also
- Almost complex manifoldAlmost complex manifoldIn mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost...
- Hyper-Kähler manifold
- Kähler–Einstein metric
- Quaternion-Kähler manifoldQuaternion-Kähler manifoldIn differential geometry, a quaternion-Kähler manifold is a Riemannian manifold whose Riemannian holonomy group is a subgroup of Sp·Sp....
- Complex Poisson manifold