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 is the intersection of any two groups among and .
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 (and hence a Riemannian metric on )., Definition 16.1. The odd-dimensional counterpart for contact manifolds is a Sasakian manifold.Charles P. Boyer, Krzysztof Galicki, "3-Sasakian Manifolds", Surveys Diff. Geom. 7 (1999) 123-184
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 (where is the complex number ). For a Kähler manifold , the Kähler form is a real closed (1,1)-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 ., Proposition 7.14. 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 ., v. 2, p. 149. Equivalently, there is a complex structure on the tangent space of at each point (that is, a real linear map from to itself with ) such that preserves the metric (meaning that ) and is preserved by parallel transport.
The symplectic form is then defined by , which is closed since is preserved by parallel transport.
Examples Complex space with the standard Hermitian metric is a Kähler manifold. A compact complex torus with a full lattice inherits a flat metric from the Euclidean metric on , 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 , which is equal to the unitary 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 , the group of linear automorphisms of that preserve the standard Hermitian form. The Fubini–Study metric is the unique Riemannian metric on (up to a positive multiple) that is invariant under the action of on . One natural generalization of 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 of a Kähler manifold is Kähler. In particular, any open subset of , any Stein manifold (embedded in ) or any smooth projective algebraic variety (embedded in ) is Kähler. This provides a large class of examples. The open unit ball in 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. Every Hermitian symmetric space of noncompact type is biholomorphic to a bounded domain in some , and the Bergman metric of is a complete Kähler metric with nonpositive sectional curvature. Every K3 surface is Kähler (by Siu).
Kähler potential A smooth real-valued function on a complex manifold is called strictly plurisubharmonic if the real closed (1,1)-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 ., Proposition 8.8. 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 On a Kähler manifold of dimension , the volume form associated with the symplectic-form is also the Riemannian volume form of the metric ., section 7.4.
A specific feature of compact Kähler manifolds is the fact that the volume of a closed complex subspace of is determined by its fundamental class viewed in . 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
where is an r-dimensional closed complex subspace and is the Kähler form. Since is closed, this integral depends only on the fundamental class of in and can be expressed as . These volumes are always positive, which expresses a strong positivity of the Kähler class in with respect to complex subspaces. In particular, for a compact Kähler manifold of complex dimension .
A related fact is that every closed complex subspace of a compact Kähler manifold 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 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 ., Section 3.1. 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. (Equivalently, is the adjoint of with respect to the L2 inner product on -forms with compact support.) 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:, Proposition 3.1.12.
These identities imply that on a Kähler manifold ,
where is the space of harmonic -forms on (forms with ) 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 , 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 of with complex coefficients splits as a direct sum of certain coherent sheaf cohomology groups:, Corollary 3.2.12.
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.
Alternatively, the Hodge decomposition can be shown to be independent of the Kähler metric by using Bott-Chern cohomology.
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 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 .