Kähler manifold
Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of view: A Kähler manifold is a symplectic manifold equipped with an integrable almost-complex structure "J" which is compatible with the symplectic form ω, meaning that the bilinear form
formula_1
on the tangent space of "X" at each point is symmetric and positive definite (and hence a Riemannian metric on "X"). 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
formula_2
for tangent vectors "u" and "v" (where "i" is the complex number formula_3). 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
formula_4
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 C"n" to order 2 near "p". That is, if the chart takes "p" to 0 in C"n", and the metric is written in these coordinates as , then
formula_5
for all "a", "b" in 
Since the 2-form "ω" is closed, it determines an element in de Rham cohomology , known as the Kähler class. A Kähler manifold is a Riemannian manifold "X" of even dimension 2"n" whose holonomy group is contained in the unitary group U("n"). 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 ) such that "J" preserves the metric "g" (meaning that ) and "J" is preserved by parallel transport. A smooth real-valued function ρ on a complex manifold is called strictly plurisubharmonic if the real closed (1,1)-form
formula_6
is positive, that is, a Kähler form. Here formula_7 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 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 formula_8. 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. 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 formula_9-lemma from Hodge theory. Namely, if formula_10 is a compact Kähler manifold, then the cohomology class formula_11 is called a Kähler class. Any other representative of this class, formula_12 say, differs from formula_13 by formula_14 for some one-form formula_15. The formula_9-lemma further states that this exact form formula_17 may be written as formula_18 for a smooth function formula_19. In the local discussion above, one takes the local Kähler class formula_20 on an open subset formula_21, and by the Poincaré lemma any Kähler form will locally be cohomologous to zero. Thus the local Kähler potential formula_22 is the same formula_23 for formula_20 locally. In general if formula_25 is a Kähler class, then any other Kähler metric can be written as formula_26 for such a smooth function. This form is not automatically a positive form, so the space of Kähler potentials for the class formula_25 is defined as those positive cases, and is commonly denoted by formula_28:
formula_29
If two Kähler potentials differ by a constant, then they define the same Kähler metric, so the space of Kähler metrics in the class formula_25 can be identified with the quotient formula_31. 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. 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
formula_32
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, "ω""n" 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 (outside its singular set). Even more: by the theory of calibrated geometry, "Y" minimizes volume among all (real) cycles in the same homology class. On a Riemannian manifold of dimension "N", the Laplacian on smooth "r"-forms is defined by
formula_33
where formula_34 is the exterior derivative and formula_35, where formula_36 is the Hodge star operator. (Equivalently, formula_37 is the adjoint of formula_34 with respect to the "L"2 inner product on "r"-forms with compact support.) For a Hermitian manifold "X", formula_34 and formula_37 are decomposed as
formula_41
and two other Laplacians are defined:
formula_42
If "X" is Kähler, then these Laplacians are all the same up to a constant:
formula_43
These identities imply that on a Kähler manifold "X", 
formula_44
where formula_45 is the space of harmonic "r"-forms on "X" (forms "α" with ) and formula_46 is the space of harmonic ("p","q")-forms. That is, a differential form formula_47 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 of "X" with complex coefficients splits as a direct sum of certain coherent sheaf cohomology groups:
formula_48
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 "H"("X") be the complex vector space , which can be identified with the space formula_49 of harmonic forms with respect to a given Kähler metric. The Hodge numbers of "X" are defined by . The Hodge decomposition implies a decomposition of the Betti numbers of a compact Kähler manifold "X" in terms of its Hodge numbers:
formula_50
The Hodge numbers of a compact Kähler manifold satisfy several identities. The Hodge symmetry holds because the Laplacian formula_51 is a real operator, and so formula_52. The identity can be proved using that the Hodge star operator gives an isomorphism formula_53. It also follows from Serre duality. A simple consequence of Hodge theory is that every odd Betti number "b"2"a"+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 "b"1 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: 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.) 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 . (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 is in .) 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 ). 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. 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. 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 "c"1("X") (the first Chern class of the tangent bundle) in . It follows that a compact Kähler–Einstein manifold "X" must have canonical bundle "K""X" 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. 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. 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 CP"n" (for ) 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 CP"n" has holomorphic sectional curvature equal to 1. At the other extreme, the open unit ball in C"n" 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 intimately related to the complex geometry of the underlying complex manifold. It is an elementary consequence of the Ahlfors Schwarz lemma that if formula_10 is a Hermitian manifold with a Hermitian metric of negative holomorphic sectional curvature (bounded above by a negative constant), then it is Brody hyperbolic (i.e., every holomorphic map formula_55 is constant). If "X" happens to be compact, then this is equivalent to the manifold being Kobayashi hyperbolic. On the other hand, if formula_10 is a compact Kähler manifold with a Kähler metric of positive holomorphic sectional curvature, Yang Xiaokui showed that "X" is rationally connected. A remarkable feature of complex geometry is that holomorphic sectional curvature decreases on complex submanifolds. (The same goes for a more general concept, holomorphic bisectional curvature.) For example, every complex submanifold of C"n" (with the induced metric from C"n") has holomorphic sectional curvature ≤ 0. 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. This was later refined by Kyle Broder, where he introduces the second Schwarz bisectional curvature, a technical refinement of the real bisectional curvature. #Complex space C"n" with the standard Hermitian metric is a Kähler manifold. #A compact complex torus C"n"/Λ (Λ a full lattice) inherits a flat metric from the Euclidean metric on C"n", 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 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. #There is a standard choice of Kähler metric on complex projective space CP"n", the Fubini–Study metric. One description involves the unitary group , the group of linear automorphisms of C"n"+1 that preserve the standard Hermitian form. The Fubini–Study metric is the unique Riemannian metric on CP"n" (up to a positive multiple) that is invariant under the action of on CP"n". One natural generalization of CP"n" 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. #The induced metric on a complex submanifold of a Kähler manifold is Kähler. In particular, any Stein manifold (embedded in C"n") or smooth projective algebraic variety (embedded in CP"n") is Kähler. This is a large class of examples. #The open unit ball B in C"n" 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 C"n", and the Bergman metric of "X" is a complete Kähler metric with sectional curvature ≤ 0. #Every K3 surface is Kähler (by Siu).
Definitions ; Symplectic viewpoint
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.