Inmathematics and especiallydifferential geometry, aKähler manifold is amanifold with three mutually compatible structures: acomplex structure, aRiemannian structure, and asymplectic structure. The concept was first studied byJan Arnoldus Schouten andDavid van Dantzig in 1930, and then introduced byErich Kähler in 1933. The terminology has been fixed byAndré 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 likeHermitian Yang–Mills connections, or special metrics such asKähler–Einstein metrics.

Everysmoothcomplexprojective variety is a Kähler manifold.Hodge theory is a central part ofalgebraic geometry, proved using Kähler metrics.

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

A Kähler manifold is asymplectic manifold${\displaystyle (X,\omega )}$ equipped with anintegrable almost-complex structure${\displaystyle J}$ which iscompatible with thesymplectic form${\displaystyle \omega }$, meaning that thebilinear form

${\displaystyle g(u,v)=\omega (u,Jv)}$

on thetangent space of${\displaystyle X}$ at each point is symmetric andpositive definite (and hence a Riemannian metric on${\displaystyle X}$).^[1]

A Kähler manifold is acomplex manifold${\displaystyle X}$ with aHermitian metric${\displaystyle h}$ whoseassociated2-form${\displaystyle \omega }$ isclosed. In more detail,${\displaystyle h}$ gives a positive definiteHermitian form on the tangent space${\displaystyle TX}$ at each point of${\displaystyle X}$, and the 2-form${\displaystyle \omega }$ is defined by

${\displaystyle \omega (u,v)=\mathrm{Re}h(iu,v)=\mathrm{Im}h(u,v)}$

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

${\displaystyle g(u,v)=\mathrm{Re}h(u,v).}$

Equivalently, a Kähler manifold${\displaystyle X}$ is aHermitian manifold of complex dimension${\displaystyle n}$ such that for every point${\displaystyle p}$ of${\displaystyle X}$, there is aholomorphiccoordinate chart around${\displaystyle p}$ in which the metric agrees with the standard metric on${\displaystyle {\mathbb{C}}^n}$ to order 2 near${\displaystyle p}$.^[2] That is, if the chart takes${\displaystyle p}$ to${\displaystyle 0}$ in${\displaystyle {\mathbb{C}}^n}$, and the metric is written in these coordinates as$h_{ab}=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{z}_a},\frac{\mathrm{\partial }}{\mathrm{\partial }{z}_b}\right)$, then

${\displaystyle h_{ab}={\delta }_{ab}+O(\Vert z{\Vert }^2)}$

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

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

A Kähler manifold is aRiemannian manifold${\displaystyle X}$ of even dimension${\displaystyle 2n}$ whoseholonomy group is contained in theunitary group${\displaystyle \mathrm{U}(n)}$.^[3] Equivalently, there is a complex structure${\displaystyle J}$ on the tangent space of${\displaystyle X}$ at each point (that is, a reallinear map from${\displaystyle TX}$ to itself with${\displaystyle J^2=-1}$) such that${\displaystyle J}$ preserves the metric${\displaystyle g}$ (meaning that${\displaystyle g(Ju,Jv)=g(u,v)}$) and${\displaystyle J}$ is preserved byparallel transport.

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

Asmooth real-valued function${\displaystyle \rho }$ on a complex manifold is calledstrictly plurisubharmonic if the real closed (1,1)-form

${\displaystyle \omega =\frac{i}{2}\mathrm{\partial }\overline{\mathrm{\partial }}\rho }$

is positive, that is, a Kähler form. Here${\displaystyle \mathrm{\partial },\overline{\mathrm{\partial }}}$ are theDolbeault operators. The function${\displaystyle \rho }$ is called aKähler potential for${\displaystyle \omega }$.

Conversely, by the complex version of thePoincaré lemma, known as thelocal${\displaystyle \mathrm{\partial }\overline{\mathrm{\partial }}}$-lemma, every Kähler metric can locally be described in this way. That is, if${\displaystyle (X,\omega )}$ is a Kähler manifold, then for every point${\displaystyle p}$ in${\displaystyle X}$ there is a neighborhood${\displaystyle U}$ of${\displaystyle p}$ and a smooth real-valued function${\displaystyle \rho }$ on${\displaystyle U}$ such that${\displaystyle {\omega |}_U=(i/2)\mathrm{\partial }\overline{\mathrm{\partial }}\rho }$.^[4] Here${\displaystyle \rho }$ is called alocal Kähler potential for${\displaystyle \omega }$. 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 formglobally using a single Kähler potential, it is possible to describe thedifference of two Kähler forms this way, provided they are in the samede Rham cohomology class. This is a consequence of the${\displaystyle \mathrm{\partial }\overline{\mathrm{\partial }}}$-lemma fromHodge theory.

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

In general if${\displaystyle [\omega ]}$ is a Kähler class, then any other Kähler metric can be written as${\displaystyle {\omega }_{\phi }=\omega +i\mathrm{\partial }\overline{\mathrm{\partial }}\phi }$ for such a smooth function. This form is not automatically apositive form, so the space ofKähler potentials for the class${\displaystyle [\omega ]}$ is defined as those positive cases, and is commonly denoted by${\displaystyle \mathcal{K}}$:

${\displaystyle {\mathcal{K}}_{[\omega ]}:=\{\phi :X\to \mathbb{R}\text{ smooth}\mid \omega +i\mathrm{\partial }\overline{\mathrm{\partial }}\phi >0\}.}$

If two Kähler potentials differ by a constant, then they define the same Kähler metric, so the space of Kähler metrics in the class${\displaystyle [\omega ]}$ can be identified with the quotient${\displaystyle \mathcal{K}/\mathbb{R}}$. The space of Kähler potentials is acontractible space. In this way the space of Kähler potentials allows one to studyall Kähler metrics in a given class simultaneously, and this perspective in the study of existence results for Kähler metrics.

For acompact Kähler manifoldX, the volume of aclosed complexsubspace ofX is determined by itshomology 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

${\displaystyle \mathrm{v}\mathrm{o}\mathrm{l}(Y)=\frac{1}{r!}{\int }_Y{\omega }^r,}$

whereY is anr-dimensional closed complex subspace andω is the Kähler form.^[5] Sinceω is closed, this integral depends only on the class ofY inH_2r(X,R). These volumes are always positive, which expresses a strong positivity of the Kähler classω inH²(X,R) with respect to complex subspaces. In particular,ωⁿ is not zero inH²ⁿ(X,R), for a compact Kähler manifoldX of complex dimensionn.

A related fact is that every closed complex subspaceY of a compact Kähler manifoldX is aminimal submanifold (outside its singular set). Even more: by the theory ofcalibrated geometry,Y minimizes volume among all (real) cycles in the same homology class.

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 thecomplex differential forms of Kähler manifolds which do not hold for arbitrary complex manifolds. These identities relate the exterior derivative${\displaystyle d}$, theDolbeault operators${\displaystyle \mathrm{\partial },\overline{\mathrm{\partial }}}$ and their adjoints, the Laplacians${\displaystyle {\mathrm{\Delta }}_d,{\mathrm{\Delta }}_{\mathrm{\partial }},{\mathrm{\Delta }}_{\overline{\mathrm{\partial }}}}$, and theLefschetz operator${\displaystyle L:=\omega \wedge -}$ and its adjoint, thecontraction operator${\displaystyle \mathrm{\Lambda }=L^{\ast }}$.^[6] 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 theKodaira andNakano vanishing theorems, theLefschetz hyperplane theorem,Hard Lefschetz theorem,Hodge-Riemann bilinear relations, andHodge index theorem.

On a Riemannian manifold of dimension${\displaystyle n}$, theLaplacian on smooth${\displaystyle r}$-forms is defined by${\displaystyle {\mathrm{\Delta }}_d=d{d}^{\ast }+d^{\ast }d}$where${\displaystyle d}$ is the exterior derivative and${\displaystyle d^{\ast }=-(-1{)}^{n(r+1)}\star d\star }$, where${\displaystyle \star }$ is theHodge star operator. (Equivalently,${\displaystyle d^{\ast }}$ is theadjoint of${\displaystyle d}$ with respect to theL² inner product on${\displaystyle r}$-forms with compact support.) For a Hermitian manifold${\displaystyle X}$,${\displaystyle d}$ and${\displaystyle d^{\ast }}$ are decomposed as

${\displaystyle d=\mathrm{\partial }+\overline{\mathrm{\partial }},d^{\ast }={\mathrm{\partial }}^{\ast }+{\overline{\mathrm{\partial }}}^{\ast },}$

and two other Laplacians are defined:

${\displaystyle {\mathrm{\Delta }}_{\overline{\mathrm{\partial }}}=\overline{\mathrm{\partial }}{\overline{\mathrm{\partial }}}^{\ast }+{\overline{\mathrm{\partial }}}^{\ast }\overline{\mathrm{\partial }},{\mathrm{\Delta }}_{\mathrm{\partial }}=\mathrm{\partial }{\mathrm{\partial }}^{\ast }+{\mathrm{\partial }}^{\ast }\mathrm{\partial }.}$

If${\displaystyle X}$ is Kähler, theKähler identities imply these Laplacians are all the same up to a constant:^[7]

${\displaystyle {\mathrm{\Delta }}_d=2{\mathrm{\Delta }}_{\overline{\mathrm{\partial }}}=2{\mathrm{\Delta }}_{\mathrm{\partial }}.}$

These identities imply that on a Kähler manifold${\displaystyle X}$,

${\displaystyle {\mathcal{H}}^r(X)=\underset{p+q=r}{⨁}{\mathcal{H}}^{p,q}(X),}$

where${\displaystyle {\mathcal{H}}^r}$ is the space ofharmonic${\displaystyle r}$-forms on${\displaystyle X}$ (forms${\displaystyle \alpha }$ with${\displaystyle \mathrm{\Delta }\alpha =0}$) and${\displaystyle {\mathcal{H}}^{p,q}}$ is the space of harmonic${\displaystyle (p,q)}$-forms. That is, a differential form${\displaystyle \alpha }$ is harmonic if and only if each of its${\displaystyle (p,q)}$-components is harmonic.

Further, for a compact Kähler manifold${\displaystyle X}$,Hodge theory gives an interpretation of the splitting above which does not depend on the choice of Kähler metric. Namely, thecohomology${\displaystyle H^r(X,\mathbf{C})}$ of${\displaystyle X}$ with complex coefficients splits as adirect sum of certaincoherent sheaf cohomology groups:^[8]

${\displaystyle H^r(X,\mathbf{C})\cong \underset{p+q=r}{⨁}{H}^q(X,{\mathrm{\Omega }}^p).}$

The group on the left depends only on${\displaystyle X}$ as a topological space, while the groups on the right depend on${\displaystyle X}$ as a complex manifold. So thisHodge decomposition theorem connects topology and complex geometry for compact Kähler manifolds.

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

${\displaystyle b_r=\sum _{p+q=r}{h}^{p,q}.}$

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

A simple consequence of Hodge theory is that every odd Betti numberb_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 theHopf surface, which isdiffeomorphic toS¹ ×S³ and hence hasb₁ = 1.

The "Kähler package" is a collection of further restrictions on the cohomology of compact Kähler manifolds, building on Hodge theory. The results include theLefschetz hyperplane theorem, thehard Lefschetz theorem, and theHodge-Riemann bilinear relations.^[9] A related result is that every compact Kähler manifold isformal in the sense of rational homotopy theory.^[10]

The question of which groups can befundamental groups of compact Kähler manifolds, calledKähler groups, is wide open. Hodge theory gives many restrictions on the possible Kähler groups.^[11] The simplest restriction is that theabelianization of a Kähler group must have even rank, since the Betti numberb₁ of a compact Kähler manifold is even. (For example, theintegersZ cannot be the fundamental group of a compact Kähler manifold.) Extensions of the theory such asnon-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 everyfinitely presented group arises as the fundamental group of some compact complex manifold of dimension 3.^[12] (Conversely, the fundamental group of anyclosed manifold is finitely presented.)

TheKodaira embedding theorem characterizes smooth complex projective varieties among all compact Kähler manifolds. Namely, a compact complex manifoldX is projective if and only if there is a Kähler formω onX whose class inH²(X,R) is in the image of the integral cohomology groupH²(X,Z). (Because a positive multiple of a Kähler form is a Kähler form, it is equivalent to say thatX has a Kähler form whose class inH²(X,R) comes fromH²(X,Q).) Equivalently,X is projective if and only if there is aholomorphic line bundleL onX with a hermitian metric whose curvature form ω is positive (since ω is then a Kähler form that represents the firstChern class ofL inH²(X,Z)). The Kähler formω that satisfies these conditions (that is, Kähler formω 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.^[13][14]

Many properties of Kähler manifolds hold in the slightly greater generality of${\displaystyle \mathrm{\partial }\overline{\mathrm{\partial }}}$-manifolds, that is compact complex manifolds for which the${\displaystyle \mathrm{\partial }\overline{\mathrm{\partial }}}$-lemma holds. In particular theBott–Chern cohomology is an alternative to theDolbeault cohomology of a compact complex manifolds, and they are isomorphic if and only if the manifold satisfies the${\displaystyle \mathrm{\partial }\overline{\mathrm{\partial }}}$-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.^[15]

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, mostcompact 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 theclassification 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 evenhomotopy equivalent to any smooth complex projective variety.^[16]

One can also ask for a characterization of compact Kähler manifolds among all compact complex manifolds. In complex dimension 2, Kodaira andYum-Tong Siu showed that a compact complex surface has a Kähler metric if and only if its first Betti number is even.^[17] 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.^[18][19] 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 calledKähler–Einstein if it has constantRicci curvature. Equivalently, the Ricci curvature tensor is equal to a constant λ times themetric tensor, Ric =λg. The reference to Einstein comes fromgeneral relativity, which asserts in the absence of mass that spacetime is a 4-dimensionalLorentzian manifold with zero Ricci curvature. See the article onEinstein 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 manifoldX can be viewed as a real closed (1,1)-form that representsc₁(X) (the first Chern class of thetangent bundle) inH²(X,R). It follows that a compact Kähler–Einstein manifoldX must havecanonical bundleK_X either anti-ample, homologically trivial, orample, depending on whether the Einstein constant λ is positive, zero, or negative. Kähler manifolds of those three types are calledFano,Calabi–Yau, or with ample canonical bundle (which impliesgeneral type), respectively. By the Kodaira embedding theorem, Fano manifolds and manifolds with ample canonical bundle are automatically projective varieties.

Shing-Tung Yau proved theCalabi 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 theMiyaoka–Yau inequality for varieties with ample canonical bundle and the Beauville–Bogomolov decomposition for Calabi–Yau manifolds.^[20]

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 isK-stable, a purely algebro-geometric condition.

In situations where there cannot exist a Kähler–Einstein metric, it is possible to study mild generalizations includingconstant scalar curvature Kähler metrics andextremal Kähler metrics. When a Kähler–Einstein metric can exist, these broader generalizations are automatically Kähler–Einstein.

The deviation of a Riemannian manifoldX from the standard metric on Euclidean space is measured bysectional curvature, which is a real number associated to any real 2-plane in the tangent space ofX at a point. For example, the sectional curvature of the standard metric onCPⁿ (forn ≥ 2) varies between 1/4 and 1 at every point. For a Hermitian manifold (for example, a Kähler manifold), theholomorphic sectional curvature means the sectional curvature restricted to complex lines in the tangent space. This behaves more simply, in thatCPⁿ has holomorphic sectional curvature equal to 1 everywhere. At the other extreme, the open unitball inCⁿ has acomplete Kähler metric with holomorphic sectional curvature equal to −1. (With this metric, the ball is also calledcomplex 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${\displaystyle (X,\omega )}$ 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${\displaystyle \mathbb{C}\to X}$ is constant). IfX happens to be compact, then this is equivalent to the manifold beingKobayashi hyperbolic.^[21]

On the other hand, if${\displaystyle (X,\omega )}$ is a compact Kähler manifold with a Kähler metric of positive holomorphic sectional curvature, Yang Xiaokui showed thatX is rationally connected.

A remarkable feature of complex geometry is that holomorphic sectional curvature decreases on complex submanifolds.^[22] (The same goes for a more general concept, holomorphic bisectional curvature.) For example, every complex submanifold ofCⁿ (with the induced metric fromCⁿ) 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 thereal bisectional curvature, introduced by Xiaokui Yang and Fangyang Zheng.^[23] This also appears in the work of Man-Chun Lee and Jeffrey Streets under the namecomplex curvature operator.^[24]

1. Complex spaceCⁿ with the standard Hermitian metric is a Kähler manifold.
2. A compact complex torusCⁿ/Λ (Λ a fulllattice) inherits a flat metric from the Euclidean metric onCⁿ, and is therefore a compact Kähler manifold.
3. Every Riemannian metric on anoriented 2-manifold is Kähler. (Indeed, its holonomy group is contained in therotation group SO(2), which is equal to the unitary group U(1).) In particular, an oriented Riemannian 2-manifold is aRiemann surface in a canonical way; this is known as the existence ofisothermal 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 oncomplex projective spaceCPⁿ, theFubini–Study metric. One description involves theunitary groupU(n + 1), the group of linear automorphisms ofCⁿ⁺¹ that preserve the standard Hermitian form. The Fubini–Study metric is the unique Riemannian metric onCPⁿ (up to a positive multiple) that is invariant under the action ofU(n + 1) onCPⁿ. One natural generalization ofCPⁿ is provided by theHermitian symmetric spaces of compact type, such asGrassmannians. The natural Kähler metric on a Hermitian symmetric space of compact type has sectional curvature ≥ 0.
5. The induced metric on acomplex submanifold of a Kähler manifold is Kähler. In particular, anyStein manifold (embedded inCⁿ) or smooth projectivealgebraic variety (embedded inCPⁿ) is Kähler. This is a large class of examples.
6. The open unit ballB inCⁿ has a complete Kähler metric called theBergman metric, with holomorphic sectional curvature equal to −1. A natural generalization of the ball is provided by theHermitian symmetric spaces of noncompact type, such as theSiegel upper half space. Every Hermitian symmetric spaceX of noncompact type is isomorphic to a bounded domain in someCⁿ, and the Bergman metric ofX is a complete Kähler metric with sectional curvature ≤ 0.
7. EveryK3 surface is Kähler (by Siu).^[17]

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

• "Kähler manifold",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Moroianu, Andrei (2004),Lectures on Kähler Geometry(PDF)

• Riemannian manifolds
• Algebraic geometry
• Complex manifolds
• Symplectic geometry

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