Kähler manifold

 ( Riemannian Manifolds ) 

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex manifold, 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 scheme complex number projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics.

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

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Symplectic viewpoint A Kähler manifold is a symplectic manifold $(X,\backslash omega)$ equipped with an integrable almost-complex structure $J$ which is compatible with the symplectic form $\backslash omega$, meaning that the bilinear form

$g(u,v)=\backslash omega(u,Jv)$

on the tangent space of $X$ at each point is symmetric and positive definite (and hence a Riemannian metric on $X$)., Definition 16.1.

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Complex viewpoint A Kähler manifold is a complex manifold $X$ with a Hermitian metric $h$ whose associated 2-form $\backslash omega$ 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 $\backslash omega$ is defined by

$\backslash omega(u,v)=\backslash operatorname\{Re\}\; h(iu,v)\; =\; \backslash operatorname\{Im\}\; h(u,\; v)$

for tangent vectors $u$ and $v$ (where $i$ is the complex number $\backslash sqrt\{-1\}$). For a Kähler manifold $X$, the Kähler form $\backslash omega$ 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

$g(u,v)=\backslash operatorname\{Re\}\; h(u,v).$

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 $\backslash mathbb\{C\}^n$ to order 2 near $p$., Proposition 7.14. That is, if the chart takes $p$ to $0$ in $\backslash mathbb\{C\}^n$, and the metric is written in these coordinates as $h\_\{ab\}=\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; z\_a\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\_b\}\backslash right)$, then

$h\_\{ab\}=\backslash delta\_\{ab\}+O(\backslash |z\backslash |^2)$

for all $a$, $b$ $\backslash in\; \backslash \{1,\backslash cdots,\; n\backslash \}$

Since the 2-form $\backslash omega$ is closed, it determines an element in de Rham cohomology $H^2(X,\backslash mathbb\{R\})$, known as the Kähler class.

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Riemannian viewpoint A Kähler manifold is a Riemannian manifold $X$ of even dimension $2n$ whose holonomy group is contained in the unitary group $\backslash operatorname\{U\}(n)$., v. 2, p. 149. 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 $J^2=-1$) such that $J$ preserves the metric $g$ (meaning that $g(Ju,Jv)=g(u,v)$) and $J$ is preserved by parallel transport.

The symplectic form $\backslash omega$ is then defined by $g(u,v)=\backslash omega(u,Jv)$, which is closed since $J$ is preserved by parallel transport.

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Kähler potential A smooth function real-valued function $\backslash rho$ on a complex manifold is called strictly plurisubharmonic if the real closed (1,1)-form

$\backslash omega\; =\; \backslash frac\; i2\; \backslash partial\; \backslash bar\backslash partial\; \backslash rho$

is positive, that is, a Kähler form. Here $\backslash partial,\; \backslash bar\backslash partial$ are the Dolbeault operators. The function $\backslash rho$ is called a Kähler potential for $\backslash omega$.

Conversely, by the complex version of the Poincaré lemma, known as the local $\backslash partial\; \backslash bar\; \backslash partial$-lemma, every Kähler metric can locally be described in this way. That is, if $(X,\backslash omega)$ is a Kähler manifold, then for every point $p$ in $X$ there is a neighborhood $U$ of $p$ and a smooth real-valued function $\backslash rho$ on $U$ such that $\{\backslash omega\backslash vert\}\_U=(i/2)\backslash partial\backslash bar\backslash partial\; \backslash rho$., Proposition 8.8. Here $\backslash rho$ is called a local Kähler potential for $\backslash omega$. There is no comparable way of describing a general Riemannian metric in terms of a single function.

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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 ddbar lemma from Hodge theory.

Namely, if $(X,\backslash omega)$ is a compact Kähler manifold, then the cohomology class $\backslash omega\backslash in\; H\_\{\backslash text\{dR\}\}^2(X)$ is called a Kähler class. Any other representative of this class, $\backslash omega\text{'}$ say, differs from $\backslash omega$ by $\backslash omega\text{'}\; =\; \backslash omega\; +\; d\backslash beta$ for some one-form $\backslash beta$. The $\backslash partial\; \backslash bar\; \backslash partial$-lemma further states that this exact form $d\backslash beta$ may be written as $d\backslash beta\; =\; i\; \backslash partial\; \backslash bar\; \backslash partial\; \backslash varphi$ for a smooth function $\backslash varphi:\; X\backslash to\; \backslash mathbb\{C\}$. In the local discussion above, one takes the local Kähler class $\backslash omega=0$ on an open subset $U\backslash subset\; X$, and by the Poincaré lemma any Kähler form will locally be cohomologous to zero. Thus the local Kähler potential $\backslash rho$ is the same $\backslash varphi$ for $\backslash omega=0$ locally.

In general if $\backslash omega$ is a Kähler class, then any other Kähler metric can be written as $\backslash omega\_\backslash varphi\; =\; \backslash omega\; +\; i\; \backslash partial\; \backslash bar\; \backslash partial\; \backslash varphi$ for such a smooth function. This form is not automatically a positive form, so the space of Kähler potentials for the class $\backslash omega$ is defined as those positive cases, and is commonly denoted by $\backslash mathcal\{K\}$:

$\backslash mathcal\{K\}\_\{\backslash omega\}\; :=\; \backslash \{\; \backslash varphi:\; X\; \backslash to\; \backslash mathbb\{R\}\; \backslash text\{\; smooth\}\; \backslash mid\; \backslash omega\; +\; i\; \backslash partial\; \backslash bar\; \backslash partial\; \backslash varphi>0\; \backslash \}.$

If two Kähler potentials differ by a constant, then they define the same Kähler metric, so the space of Kähler metrics in the class $\backslash omega$ can be identified with the quotient $\backslash mathcal\{K\}/\backslash mathbb\{R\}$. 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.

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Kähler manifolds and volume minimizers For a compact space Kähler manifold X, the volume of a closed set 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

$\backslash mathrm\{vol\}(Y)=\backslash frac\{1\}\{r!\}\backslash int\_Y\; \backslash omega^r,$

where Y is an r-dimensional closed complex subspace and ω is the Kähler form., section 7.4. 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.

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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 $d$, the Dolbeault operators $\backslash partial,\; \backslash bar\; \backslash partial$ and their adjoints, the Laplacians $\backslash Delta\_d,\; \backslash Delta\_\{\backslash partial\},\; \backslash Delta\_\{\backslash bar\; \backslash partial\}$, and the Lefschetz operator $L\; :=\; \backslash omega\; \backslash wedge\; -$ and its adjoint, the contraction operator $\backslash Lambda\; =\; L^*$., 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.

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The Laplacian on a Kähler manifold On a Riemannian manifold of dimension $n$, the Laplacian on smooth $r$-forms is defined by $\backslash Delta\_d=dd^*+d^*d$ where $d$ is the exterior derivative and $d^*=-(-1)^\{n(r+1)\}\backslash star\; d\backslash ,\; \backslash star$, where $\backslash star$ is the Hodge star operator. (Equivalently, $d^*$ is the adjoint operator of $d$ with respect to the L² inner product on $r$-forms with compact support.) For a Hermitian manifold $X$, $d$ and $d^*$ are decomposed as

$d=\backslash partial+\backslash bar\{\backslash partial\},\backslash \; \backslash \; \backslash \; \backslash \; d^*=\backslash partial^*+\backslash bar\{\backslash partial\}^*,$

and two other Laplacians are defined:

$\backslash Delta\_\{\backslash bar\{\backslash partial\}\}=\backslash bar\{\backslash partial\}\backslash bar\{\backslash partial\}^*+\backslash bar\{\backslash partial\}^*\backslash bar\{\backslash partial\},\backslash \; \backslash \; \backslash \; \backslash \; \backslash Delta\_\backslash partial=\backslash partial\backslash partial^*+\backslash partial^*\backslash partial.$

If $X$ is Kähler, the Kähler identities imply these Laplacians are all the same up to a constant:, Proposition 3.1.12.

$\backslash Delta\_d=2\backslash Delta\_\{\backslash bar\{\backslash partial\}\}=2\backslash Delta\_\backslash partial\; .$

These identities imply that on a Kähler manifold $X$,

$\backslash mathcal\; H^r(X)=\backslash bigoplus\_\{p+q=r\}\backslash mathcal\; H^\{p,q\}(X),$

where $\backslash mathcal\; H^r$ is the space of harmonic $r$-forms on $X$ (forms $\backslash alpha$ with $\backslash Delta\backslash alpha=0$) and $\backslash mathcal\; H^\{p,q\}$ is the space of harmonic $(p,q)$-forms. That is, a differential form $\backslash alpha$ 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 $H^r(X,\backslash mathbf\{C\})$ of $X$ with complex coefficients splits as a direct sum of certain coherent sheaf cohomology groups:, Corollary 3.2.12.

$H^r(X,\backslash mathbf\{C\})\backslash cong\backslash bigoplus\_\{p+q=r\}H^q(X,\backslash Omega^p).$

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 $H^\{p,q\}(X)$ be the complex vector space $H^q(X,\backslash Omega^p)$, which can be identified with the space $\backslash mathcal\; H^\{p,q\}(X)$ of harmonic forms with respect to a given Kähler metric. The Hodge numbers of $X$ are defined by $h^\{p,q\}(X)=\backslash mathrm\{dim\}\_\{\backslash mathbf\{C\}\}H^\{p,q\}(X)$. The Hodge decomposition implies a decomposition of the of a compact Kähler manifold $X$ in terms of its Hodge numbers:

$b\_r=\backslash sum\_\{p+q=r\}h^\{p,q\}.$

The Hodge numbers of a compact Kähler manifold satisfy several identities. The Hodge symmetry $h^\{p,q\}=h^\{q,p\}$ holds because the Laplacian $\backslash Delta\_d$ is a real operator, and so $H^\{p,q\}=\backslash overline\{H^\{q,p\}\}$. The identity $h^\{p,q\}=h^\{n-p,n-q\}$ can be proved using that the Hodge star operator gives an isomorphism $H^\{p,q\}\backslash cong\; \backslash overline\{H^\{n-p,n-q\}\}$. It also follows from Serre duality.

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Topology of compact Kähler manifolds 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., sections 3.3 and 5.2, A related result is that every compact Kähler manifold is formal in the sense of rational homotopy theory., Proposition 3.A.28.

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. (For example, the 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., Corollary 1.66. (Conversely, the fundamental group of any closed manifold is finitely presented.)

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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 . (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 comes from .) 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 ). The Kähler form ω that satisfies these conditions (that is, Kähler form ω is an integral differential form) 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. p.217 Definition 1.1

Many properties of Kähler manifolds hold in the slightly greater generality of $\backslash partial\; \backslash bar\; \backslash partial$-manifolds, that is compact complex manifolds for which the Ddbar 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 $\backslash partial\; \backslash bar\; \backslash partial$-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.Angella, D. and Tomassini, A., 2013. On the $\backslash partial\backslash overline\; \{\backslash partial\}$-Lemma and Bott-Chern cohomology. Inventiones mathematicae, 192(1), pp.71-81.

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 complex torus 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., section IV.3. 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.

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Kähler–Einstein manifolds 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 . 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 variety, 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., Corollary 9.8.

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–Gang 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.

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Holomorphic sectional curvature 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 at every point. 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 everywhere. 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 $(X,\backslash omega)$ 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 $\backslash mathbb\{C\}\backslash to\; X$ is constant). If X happens to be compact, then this is equivalent to the manifold being Kobayashi metric., Lemma 9.14.

On the other hand, if $(X,\backslash omega)$ 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., v. 2, Proposition IX.9.2. (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.

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Examples

1. Complex space C ⁿ with the standard Hermitian metric is a Kähler manifold.
2. 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.
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 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.
4. 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 . 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 C ⁿ) or smooth projective algebraic variety (embedded in CP ⁿ) is Kähler. This is a large class of examples.
6. 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.
7. Every K3 surface is Kähler (by Siu).

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See also

• Almost complex manifold
• Hyperkähler manifold
• Quaternion-Kähler manifold
• K-energy functional

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Notes

• (1994). 9780471050599, John Wiley & Sons. ISBN 9780471050599
• (2025). 9780387738925, Springer. ISBN 9780387738925

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