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, such as the existence of special connections like 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:

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 of 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. In more detail, gives a positive definite Hermitian form on the tangent space at each point of, and the 2-form is defined by
for tangent vectors and . For a Kähler manifold, the Kähler form is a real closed -form. A Kähler manifold can also be viewed as a Riemannian manifold, with the Riemannian metric defined by
Equivalently, a Kähler manifold is a Hermitian manifold of complex dimension such that for every point of, there is a holomorphic coordinate chart around in which the metric agrees with the standard metric on to order 2 near. That is, if the chart takes to in, and the metric is written in these coordinates as, then
for all,
Since the 2-form is closed, it determines an element in de Rham cohomology, known as the Kähler class.

Riemannian viewpoint

A Kähler manifold is a Riemannian manifold of even dimension whose holonomy group is contained in the unitary group. Equivalently, there is a complex structure on the tangent space of at each point such that preserves the metric and is preserved by parallel transport.
The symplectic form is then defined by, which is closed since is preserved by parallel transport.

Kähler potential

A smooth real-valued function on a complex manifold is called strictly plurisubharmonic if the real closed -form
is positive, that is, a Kähler form. Here are the Dolbeault operators. The function is called a Kähler potential for.
Conversely, by the complex version of the Poincaré lemma, known as the local -lemma, every Kähler metric can locally be described in this way. That is, if is a Kähler manifold, then for every point in there is a neighborhood of and a smooth real-valued function on such that. Here is called a local Kähler potential for. 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 -lemma from Hodge theory.
Namely, if is a compact Kähler manifold, then the cohomology class is called a Kähler class. Any other representative of this class, say, differs from by for some one-form. The -lemma further states that this exact form may be written as for a smooth function. In the local discussion above, one takes the local Kähler class on an open subset, and by the Poincaré lemma any Kähler form will locally be cohomologous to zero. Thus the local Kähler potential is the same for locally.
In general if is a Kähler class, then any other Kähler metric can be written as for such a smooth function. This form is not automatically a positive form, so the space of Kähler potentials for the class is defined as those positive cases, and is commonly denoted by :
If two Kähler potentials differ by a constant, then they define the same Kähler metric, so the space of Kähler metrics in the class can be identified with the quotient. 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

For a compact Kähler manifold X, the volume of a closed complex subspace of X is determined by its homology class. In a sense, this means that the geometry of a complex subspace is bounded in terms of its topology. Explicitly, Wirtinger's formula says that
where Y is an r-dimensional closed complex subspace and ω is the Kähler form. Since ω is closed, this integral depends only on the class of Y in. These volumes are always positive, which expresses a strong positivity of the Kähler class ω in with respect to complex subspaces. In particular, ωⁿ is not zero in, for a compact Kähler manifold X of complex dimension n.
A related fact is that every closed complex subspace Y of a compact Kähler manifold X is a minimal submanifold. Even more: by the theory of calibrated geometry, Y minimizes volume among all 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 of Kähler manifolds which do not hold for arbitrary complex manifolds. These identities relate the exterior derivative, the Dolbeault operators and their adjoints, the Laplacians, 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

On a Riemannian manifold of dimension, the Laplacian on smooth -forms is defined by
where is the exterior derivative and, where is the Hodge star operator. For a Hermitian manifold, and are decomposed as
and two other Laplacians are defined:
If is Kähler, the Kähler identities imply these Laplacians are all the same up to a constant:
These identities imply that on a Kähler manifold,
where is the space of harmonic -forms on and is the space of harmonic -forms. That is, a differential form is harmonic if and only if each of its -components is harmonic.
Further, for a compact Kähler manifold, Hodge theory gives an interpretation of the splitting above which does not depend on the choice of Kähler metric. Namely, the cohomology of with complex coefficients splits as a direct sum of certain coherent sheaf cohomology groups:
The group on the left depends only on as a topological space, while the groups on the right depend on as a complex manifold. So this Hodge decomposition theorem connects topology and complex geometry for compact Kähler manifolds.
Let be the complex vector space, which can be identified with the space of harmonic forms with respect to a given Kähler metric. The Hodge numbers of are defined by. The Hodge decomposition implies a decomposition of the Betti numbers of a compact Kähler manifold in terms of its Hodge numbers:
The Hodge numbers of a compact Kähler manifold satisfy several identities. The Hodge symmetry holds because the Laplacian is a real operator, and so. The identity can be proved using that the Hodge star operator gives an isomorphism. It also follows from Serre duality.

Topology of compact Kähler manifolds

A simple consequence of Hodge theory is that every odd Betti number b_2a+1 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 to and hence has.
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 Kähler group must have even rank, since the Betti number b₁ of a compact Kähler manifold is even. 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.

Characterizations of complex projective varieties and compact Kähler manifolds

The Kodaira embedding theorem characterizes smooth complex projective varieties among all compact Kähler manifolds. Namely, a compact complex manifold X is projective if and only if there is a Kähler form ω on X whose class in is in the image of the integral cohomology group. Equivalently, X is projective if and only if there is a holomorphic line bundle L on X with a hermitian metric whose curvature form ω is positive. The Kähler form ω that satisfies these conditions is also called the Hodge form, and the Kähler metric at this time is called the Hodge metric. The compact Kähler manifolds with Hodge metric are also called Hodge manifolds.
Many properties of Kähler manifolds hold in the slightly greater generality of -manifolds, that is compact complex manifolds for which the -lemma holds. In particular the Bott–Chern cohomology is an alternative to the Dolbeault cohomology of a compact complex manifolds, and they are isomorphic if and only if the manifold satisfies the -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 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, but one fiber is not Kähler. Thus a compact Kähler manifold can be diffeomorphic to a non-Kähler complex manifold.