Kähler manifold

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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 introduced by Erich Kähler in 1933.

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

Definitions[edit]

Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of view:

Symplectic viewpoint[edit]

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

on the tangent space of X at each point is symmetric and positive definite (and hence a Riemannian metric on X).[1]

Complex viewpoint[edit]

A Kähler manifold is a complex manifold X with a Hermitian metric h whose associated 2-form ω is closed. In more detail, h gives a positive definite Hermitian form on the tangent space TX at each point of X, and the 2-form ω is defined by

for tangent vectors u and v (where i is the complex number ). For a Kähler manifold X, the Kähler form ω is a real closed (1,1)-form. A Kähler manifold can also be viewed as a Riemannian manifold, with the Riemannian metric g defined by

Equivalently, a Kähler manifold X is a Hermitian manifold of complex dimension n such that for every point p of X, there is a holomorphic coordinate chart around p in which the metric agrees with the standard metric on Cn to order 2 near p.[2] That is, if the chart takes p to 0 in Cn, and the metric is written in these coordinates as hab=⟨∂/∂za,∂/∂zb), then

for all a,b in {1,...,n}.

Since the 2-form ω is closed, it determines an element in de Rham cohomology H2(X,R), known as the Kähler class.

Riemannian viewpoint[edit]

A Kähler manifold is a Riemannian manifold X of even dimension 2n whose holonomy group is contained in the unitary group U(n).[3] Equivalently, there is a complex structure J on the tangent space of X at each point (that is, a real linear map from TX to itself with J2 = −1) such that J preserves the metric g (meaning that g(Ju,Jv) = g(u,v)) and J is preserved by parallel transport.

Kähler potential[edit]

A smooth real-valued function ρ 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 ρ is called a Kähler potential for ω.

Conversely, by the complex version of the Poincaré lemma, every Kähler metric can locally be described in this way. That is, if (X,ω) is a Kähler manifold, then for every point p in X there is a neighborhood U of p and a smooth real-valued function ρ on U such that .[4] Here ρ is called a local Kähler potential for ω. There is no comparable way of describing a general Riemannian metric in terms of a single function.

Kähler manifolds and volume minimizers[edit]

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. (This fails completely for real submanifolds.) Explicitly, Wirtinger's formula says that

where Y is an r-dimensional closed complex subspace and ω is the Kähler form.[5] Since ω is closed, this integral depends only on the class of Y in H2r(X,R). These volumes are always positive, which expresses a strong positivity of the Kähler class ω in H2(X,R) with respect to complex subspaces. In particular, ωn is not zero in H2n(X,R), 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 (outside its singular set). Even more: by the theory of calibrated geometry, Y minimizes volume among all (real) cycles in the same homology class.

The Laplacian on a Kähler manifold[edit]

On a Riemannian manifold of dimension N, the Laplacian on smooth r-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 r-forms with compact support.) For a Hermitian manifold X, and are decomposed as

and two other Laplacians are defined:

If X is Kähler, then these Laplacians are all the same up to a constant:[6]

These identities imply that on a Kähler manifold X,

where is the space of harmonic r-forms on X (forms α with Δα = 0) and is the space of harmonic (p,q)-forms. That is, a differential form is harmonic if and only if each of its (p,q)-components is harmonic.

Further, for a compact Kähler manifold X, Hodge theory gives an interpretation of the splitting above which does not depend on the choice of Kähler metric. Namely, the cohomology Hr(X,C) of X with complex coefficients splits as a direct sum of certain coherent sheaf cohomology groups:[7]

The group on the left depends only on X as a topological space, while the groups on the right depend on X as a complex manifold. So this Hodge decomposition theorem connects topology and complex geometry for compact Kähler manifolds.

Let Hp,q(X) be the complex vector space Hq(Xq), which can be identified with the space of harmonic forms with respect to a given Kähler metric. The Hodge numbers of X are defined by hp,q(X)=dimCHp,q(X). The Hodge decomposition implies a decomposition of the Betti numbers of a compact Kähler manifold X in terms of its Hodge numbers:

The Hodge numbers of a compact Kähler manifold satisfy several identities. The Hodge symmetry hp,q = hq,p holds because the Laplacian is a real operator, and so . The identity hp,q = hnp,nq can be proved using that the Hodge star operator gives an isomorphism . It also follows from Serre duality.

Topology of compact Kähler manifolds[edit]

A simple consequence of Hodge theory is that every odd Betti number b2a+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 S1 × S3 and hence has b1 = 1.

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.[8] A related result is that every compact Kähler manifold is formal in the sense of rational homotopy theory.[9]

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.[10] 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. (For example, the integers Z cannot be 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: Taubes showed that every finitely presented group arises as the fundamental group of some compact complex manifold of dimension 3.[11] (Conversely, the fundamental group of any closed manifold is finitely presented.)

Characterizations of complex projective varieties and compact Kähler manifolds[edit]

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 H2(X,R) is in the image of the integral cohomology group H2(X,Z). (Because a positive multiple of a Kähler form is a Kähler form, it is equivalent to say that X has a Kähler form whose class in H2(X,R) is in H2(X,Q).) 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 (since ω is then a Kähler form that represents the first Chern class of L in H2(X,Z)).

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. 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. 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.[12]

One can also ask for a characterization of compact Kähler manifolds among all compact complex manifolds. In complex dimension 2, Kodaira and Siu showed that a compact complex surface has a Kähler metric if and only if its first Betti number is even.[13] 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[edit]

A Kähler manifold is called Kähler–Einstein if it has constant Ricci curvature. Equivalently, the Ricci curvature tensor is equal to a constant λ times the metric tensor, Ric = λg. 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.

Although Ricci curvature is defined for any Riemannian manifold, it plays a special role in Kähler geometry: the Ricci curvature of a Kähler manifold X can be viewed as a real closed (1,1)-form that represents c1(X) (the first Chern class of the tangent bundle) in H2(X,R). It follows that a compact Kähler–Einstein manifold X must have canonical bundle KX either anti-ample, homologically trivial, or ample, depending on whether the Einstein constant λ is positive, zero, or negative. 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.

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.[14]

By contrast, not every smooth Fano variety has a Kähler–Einstein metric (which would have constant positive Ricci curvature). However, Chen–Donaldson–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.

Holomorphic sectional curvature[edit]

The deviation of a Riemannian manifold X from the standard metric on Euclidean space is measured by sectional curvature, which is a real number associated to any real 2-plane in the tangent space of X at a point. For example, the sectional curvature of the standard metric on CPn (for n ≥ 2) varies between 1/4 and 1. For a Hermitian manifold (for example, a Kähler manifold), the holomorphic sectional curvature means the sectional curvature restricted to complex lines in the tangent space. This behaves more simply, in that CPn has holomorphic sectional curvature equal to 1. At the other extreme, the open unit ball in Cn has a complete Kähler metric with holomorphic sectional curvature equal to −1. (With this metric, the ball is also called complex hyperbolic space.)

The holomorphic sectional curvature is closely tied to the properties of X as a complex manifold. For example, every Hermitian manifold X with holomorphic sectional curvature bounded above by a negative constant is Kobayashi hyperbolic.[15] It follows that every holomorphic map CX is constant.

A remarkable feature of complex geometry is that holomorphic sectional curvature decreases on complex submanifolds.[16] (The same goes for a more general concept, holomorphic bisectional curvature.) For example, every complex submanifold of Cn (with the induced metric from Cn) has holomorphic sectional curvature ≤ 0.

Examples[edit]

  1. Complex Euclidean space Cn with the standard Hermitian metric is a Kähler manifold.
  2. A compact complex torus Cn/Λ (Λ a full lattice) inherits a flat metric from the Euclidean metric on Cn, and is therefore a compact Kähler manifold.
  3. Every Riemannian metric on an oriented 2-manifold is Kähler. (Indeed, its holonomy group is contained in the rotation group SO(2), which is equal to the unitary group U(1).) In particular, an oriented Riemannian 2-manifold is a complex curve in a canonical way; this is known as the existence of isothermal coordinates.
  4. There is a standard choice of Kähler metric on complex projective space CPn, the Fubini–Study metric. One description involves the unitary group U(n+1), the group of linear automorphisms of Cn+1 that preserve the standard Hermitian form. The Fubini–Study metric is the unique Riemannian metric on CPn (up to a positive multiple) that is invariant under the action of U(n+1) on CPn. One natural generalization of CPn 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 sectional curvature ≥ 0.
  5. The induced metric on a complex submanifold of a Kähler manifold is Kähler. In particular, any Stein manifold (embedded in Cn) or smooth projective algebraic variety (embedded in CPn) is Kähler. This is a large class of examples.
  6. The open unit ball B in Cn 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 X of noncompact type is isomorphic to a bounded domain in some Cn, and the Bergman metric of X is a complete Kähler metric with sectional curvature ≤ 0.
  7. Every K3 surface is Kähler (by Siu).[13]

See also[edit]

Notes[edit]

  1. ^ Cannas da Silva (2001), Definition 16.1,
  2. ^ Zheng (2000), Proposition 7.14.
  3. ^ Kobayashi & Nomizu (1996), v. 2, p. 149.
  4. ^ Moroianu (2007), Proposition 8.8.
  5. ^ Zheng (2000), section 7.4.
  6. ^ Huybrechts (2005), Proposition 3.1.12.
  7. ^ Huybrechts (2005), Corollary 3.2.12.
  8. ^ Huybrechts (2005), sections 3.3 and 5.2,
  9. ^ Huybrechts (2005), Proposition 3.A.28.
  10. ^ Amorós et al. (1996).
  11. ^ Amorós et al. (1996), Corollary 1.66.
  12. ^ Voisin (2004).
  13. ^ a b Barth et al. (2004), section IV.3.
  14. ^ Zheng (2000), Corollary 9.8.
  15. ^ Zheng (2000), Lemma 9.14.
  16. ^ Kobayashi & Nomizu (1996), v. 2, Proposition IX.9.2.

References[edit]

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