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Wikipedia / Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, for example the existence of special connections such as Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics.

Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics.

Definitions

Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of view. The equivalence of these points of view essentially comes from the fact that the unitary group.

Symplectic viewpoint

A Kähler manifold is a symplectic manifold equipped with an integrable almost-complex structure which is compatible with the symplectic form, meaning that the bilinear form

on the tangent space at each point is symmetric and positive definite).

Complex viewpoint

A Kähler manifold is a complex manifold with a Hermitian metric whose associated 2-form is closed gives a positive definite Hermitian form is defined by

, the Kähler form is a real closed (1,1)-form defined by

is a Hermitian manifold, there is a holomorphic coordinate chart., then

is closed, it determines an element in de Rham cohomology, known as the Kähler class.

Riemannian viewpoint

A Kähler manifold is a Riemannian manifold whose holonomy group is contained in the unitary group. at each point (that is, a real linear map is preserved by parallel transport.

is preserved by parallel transport.

Examples

Complex space with the standard Hermitian metric is a Kähler manifold.
a full lattice, and is therefore a compact Kähler manifold.
Every Riemannian metric on an oriented 2-manifold is Kähler. (Indeed, its holonomy group is contained in the rotation group.) In particular, an oriented Riemannian 2-manifold is a Riemann surface in a canonical way; this is known as the existence of isothermal coordinates. Conversely, every Riemann surface is Kähler since the Kähler form of any Hermitian metric is closed for dimensional reasons.
There is a standard choice of Kähler metric on complex projective space, the Fubini–Study metric. One description involves the unitary group is provided by the Hermitian symmetric spaces of compact type, such as Grassmannians. The natural Kähler metric on a Hermitian symmetric space of compact type has nonnegative sectional curvature.
The induced metric on a complex submanifold, any Stein manifold) or any smooth projective algebraic variety) is Kähler. This provides a large class of examples.
has a complete Kähler metric called the Bergman metric, with holomorphic sectional curvature equal to −1. A natural generalization of the ball is provided by the Hermitian symmetric spaces of noncompact type, such as the Siegel upper half space is a complete Kähler metric with nonpositive sectional curvature.
Every K3 surface is Kähler (by Siu).

Kähler potential

A smooth on a complex manifold is called strictly plurisubharmonic if the real closed (1,1)-form

are the Dolbeault operators is called a Kähler potential.

Conversely, by the complex version of the Poincaré lemma, known as the \partial \bar \partial-lemma. is called a local Kähler potential. There is no comparable way of describing a general Riemannian metric in terms of a single function.

Space of Kähler potentials

Whilst it is not always possible to describe a Kähler form globally using a single Kähler potential, it is possible to describe the difference of two Kähler forms this way, provided they are in the same de Rham cohomology class. This is a consequence of the \partial \bar \partial-lemma from Hodge theory.

is called a Kähler class locally.

for such a smooth function. This form is not automatically a positive form, so the space of Kähler potentials:

. The space of Kähler potentials is a contractible space. In this way the space of Kähler potentials allows one to study all Kähler metrics in a given class simultaneously, and this perspective in the study of existence results for Kähler metrics.

Kähler manifolds and volume minimizers

, the volume form.

A specific feature of compact Kähler manifolds is the fact that the volume of a closed complex subspace is determined by its fundamental class. In a sense, this means that the geometry of a complex subspace is bounded in terms of its topology. (This fails completely for real submanifolds.) Explicitly, Wirtinger's formula says that

is an r is the Kähler form. and can be expressed as

is a minimal submanifold (outside its singular set). Even more: by the theory of calibrated geometry minimizes volume among all (real) cycles in the same homology class.

Kähler identities

As a consequence of the strong interaction between the smooth, complex, and Riemannian structures on a Kähler manifold, there are natural identities between the various operators on the complex differential forms, the Dolbeault operators, and the Lefschetz operator and its adjoint, the contraction operator. The identities form the basis of the analytical toolkit on Kähler manifolds, and combined with Hodge theory are fundamental in proving many important properties of Kähler manifolds and their cohomology. In particular the Kähler identities are critical in proving the Kodaira and Nakano vanishing theorems, the Lefschetz hyperplane theorem, Hard Lefschetz theorem, Hodge-Riemann bilinear relations, and Hodge index theorem.

The Laplacian on a Kähler manifold

, the Laplacian is the Hodge star operator is the adjoint with respect to the L2 inner product are decomposed as

and two other Laplacians are defined:

is Kähler, the Kähler identities imply these Laplacians are all the same up to a constant:

is the space of harmonic is the space of harmonic (p,q)-forms-components is harmonic.

, every cohomology class in de Rham or in Dolbeault cohomology contains a unique harmonic representant. The decomposition of harmonic forms into (p,q)-components therefore gives rise to the Hodge decomposition

Hodge theory gives moreover an interpretation of this splitting which does not depend on the choice of Kähler metric. Namely, the cohomology with complex coefficients splits as a direct sum of certain coherent sheaf cohomology groups:

as a complex manifold. So this Hodge decomposition theorem connects topology and complex geometry for compact Kähler manifolds.

Alternatively, the Hodge decomposition can be shown to be independent of the Kähler metric by using Bott-Chern cohomology.

of harmonic forms with respect to a given Kähler metric. The Hodge numbers. The Hodge decomposition implies a decomposition of the Betti numbers in terms of its Hodge numbers:

The Hodge numbers of a compact Kähler manifold satisfy several identities. The Hodge symmetry. It also follows from Serre duality.

Topology of compact Kähler manifolds

of a compact Kähler manifold is even, by Hodge symmetry. This is not true for compact complex manifolds in general, as shown by the example of the Hopf surface, which is diffeomorphic.

The "Kähler package" is a collection of further restrictions on the cohomology of compact Kähler manifolds, building on Hodge theory. The results include the Lefschetz hyperplane theorem, the hard Lefschetz theorem, and the Hodge-Riemann bilinear relations. A related result is that every compact Kähler manifold is formal in the sense of rational homotopy theory.

The question of which groups can be fundamental groups of compact Kähler manifolds, called Kähler groups, is wide open. Hodge theory gives many restrictions on the possible Kähler groups. The simplest restriction is that the abelianization of a compact Kähler manifold is even. (For example, the integers is not the fundamental group of a compact Kähler manifold.) Extensions of the theory such as non-abelian Hodge theory give further restrictions on which groups can be Kähler groups.

Without the Kähler condition, the situation is simple: Clifford Taubes showed that every finitely presented group arises as the fundamental group of some compact complex manifold of dimension 3. (Conversely, the fundamental group of any closed manifold is finitely presented.)

Characterizations of complex projective varieties and compact Kähler manifolds

The Kodaira embedding theorem is projective if and only if there is a holomorphic line bundle is then a Kähler form that represents the first Chern class is called a Hodge metric. A compact Kähler manifold endowed with a Hodge metric is called a Hodge manifold.

-manifolds, that is compact complex manifolds for which the \partial \bar \partial-lemma holds. In particular the Bott–Chern cohomology is an alternative to the Dolbeault cohomology-lemma, and in particular agree when the manifold is Kähler. In general the kernel of the natural map from Bott–Chern cohomology to Dolbeault cohomology contains information about the failure of the manifold to be Kähler.

Every compact complex curve is projective, but in complex dimension at least 2, there are many compact Kähler manifolds that are not projective; for example, most compact complex tori are not projective. One may ask whether every compact Kähler manifold can at least be deformed (by continuously varying the complex structure) to a smooth projective variety. Kunihiko Kodaira's work on the classification of surfaces implies that every compact Kähler manifold of complex dimension 2 can indeed be deformed to a smooth projective variety. Claire Voisin found, however, that this fails in dimensions at least 4. She constructed a compact Kähler manifold of complex dimension 4 that is not even homotopy equivalent to any smooth complex projective variety.

One can also ask for a characterization of compact Kähler manifolds among all compact complex manifolds. In complex dimension 2, Kodaira and Yum-Tong Siu showed that a compact complex surface has a Kähler metric if and only if its first Betti number is even. An alternative proof of this result which does not require the hard case-by-case study using the classification of compact complex surfaces was provided independently by Buchdahl and Lamari. Thus "Kähler" is a purely topological property for compact complex surfaces. Hironaka's example shows, however, that this fails in dimensions at least 3. In more detail, the example is a 1-parameter family of smooth compact complex 3-folds such that most fibers are Kähler (and even projective), but one fiber is not Kähler. Thus a compact Kähler manifold can be diffeomorphic to a non-Kähler complex manifold.

Kähler–Einstein manifolds

A Kähler manifold is called Kähler–Einstein if it has constant Ricci curvature times the metric tensor. The reference to Einstein comes from general relativity, which asserts in the absence of mass that spacetime is a 4-dimensional Lorentzian manifold with zero Ricci curvature. See the article on Einstein manifolds for more details.

can be viewed as a real closed (1,1)-form that represents the first Chern class of the tangent bundle is either anti-ample, homologically trivial, or ample is positive, zero, or negative. Compact Kähler manifolds of those three types are called Fano, Calabi–Yau, or with ample canonical bundle (which implies general type), respectively. By the Kodaira embedding theorem, Fano manifolds and manifolds with ample canonical bundle are automatically projective varieties.

Shing-Tung Yau proved the Calabi conjecture: every smooth projective variety with ample canonical bundle has a Kähler–Einstein metric (with constant negative Ricci curvature), and every Calabi–Yau manifold has a Kähler–Einstein metric (with zero Ricci curvature). These results are important for the classification of algebraic varieties, with applications such as the Miyaoka–Yau inequality for varieties with ample canonical bundle and the Beauville–Bogomolov decomposition for Calabi–Yau manifolds.

By contrast, not every smooth Fano variety has a Kähler–Einstein metric (which would have constant positive Ricci curvature). However, Xiuxiong Chen, Simon Donaldson, and Song Sun proved the Yau–Tian–Donaldson conjecture: a smooth Fano variety has a Kähler–Einstein metric if and only if it is K-stable, a purely algebro-geometric condition.

In situations where there cannot exist a Kähler–Einstein metric, it is possible to study mild generalizations including constant scalar curvature Kähler metrics and extremal Kähler metrics. When a Kähler–Einstein metric can exist, these broader generalizations are automatically Kähler–Einstein.

Kähler manifolds of constant curvature

Holomorphic sectional curvature

from the standard metric on Euclidean space is measured by sectional curvature at a point. For a Hermitian manifold (for example, a Kähler manifold), the holomorphic sectional curvature is defined as the sectional curvature restricted to complex lines in the tangent space.

On a Kähler manifold, the holomorphic sectional curvature completely determines the sectional curvature, and hence the curvature tensor.

happens to be compact, then this is equivalent to the manifold being Kobayashi hyperbolic.

is rationally connected.

A remarkable feature of complex geometry is that holomorphic sectional curvature decreases on complex submanifolds.) has nonpositive holomorphic sectional curvature.

For holomorphic maps between Hermitian manifolds, the holomorphic sectional curvature is not strong enough to control the target curvature term appearing in the Schwarz lemma second-order estimate. This motivated the consideration of the real bisectional curvature, introduced by Xiaokui Yang and Fangyang Zheng. This also appears in the work of Man-Chun Lee and Jeffrey Streets under the name complex curvature operator.

Kähler manifolds of constant holomorphic sectional curvature

) varies between 1/4 and 1 at every point, while this space has constant holomorphic sectional curvature equal to 1.
, the Euclidean metric is a Kähler metric of constant Riemannian, and hence holomorphic sectional curvature, equal to 0.
On the open unit ball, there is a complete. Up to a constant, this metric coincides with the Bergman metric, the Koyabashi metric, and it makes the ball a Hermitian symmetric space of noncompact type.)

As a consequence, any complete Kähler manifold of constant holomorphic sectional curvature is holomorphically isometric to a non-singular quotient of one of these 3 model spaces.