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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and especially
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a Kähler manifold is a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
with three mutually compatible structures: a complex structure, a Riemannian structure, and a
symplectic structure Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
. 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 André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is du ...
. 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 metric In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The ...
s. Every smooth
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
projective variety In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...
is a Kähler manifold.
Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
is a central part of
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, proved using Kähler metrics.


Definitions

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


Symplectic viewpoint

A Kähler manifold is a
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
(X,\omega) equipped with an integrable almost-complex structure J which is compatible with the
symplectic form In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping \omega : V \times V \to F that is ; Bilinear: ...
\omega, meaning that the
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
:g(u,v)=\omega(u,Jv) on the
tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
of X at each point is symmetric and positive definite (and hence a Riemannian metric on X).


Complex viewpoint

A Kähler manifold is a
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
X with a Hermitian metric h whose associated 2-form \omega is closed. In more detail, h gives a positive definite
Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear map, linear in each of its arguments, but a sesquilinear f ...
on the tangent space TX at each point of X, and the 2-form \omega is defined by :\omega(u,v)=\operatorname h(iu,v) = \operatorname h(u, v) for tangent vectors u and v (where i is the complex number \sqrt). For a Kähler manifold X, the Kähler form \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)=\operatorname 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 In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...
coordinate chart around p in which the metric agrees with the standard metric on \mathbb^n to order 2 near p. That is, if the chart takes p to 0 in \mathbb^n, and the metric is written in these coordinates as h_=\left(\frac, \frac\right), then :h_=\delta_+O(\, z\, ^2) for all a, b \in \ Since the 2-form \omega is closed, it determines an element in
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
H^2(X,\mathbb), known as the Kähler class.


Riemannian viewpoint

A Kähler manifold is a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
X of even dimension 2n whose holonomy group is contained in the
unitary group Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semi ...
\operatorname(n). Equivalently, there is a complex structure J on the tangent space of X at each point (that is, a real
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
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 In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on ...
.


Kähler potential

A smooth real-valued function \rho on a complex manifold is called strictly plurisubharmonic if the real closed (1,1)-form : \omega = \frac i2 \partial \bar\partial \rho is positive, that is, a Kähler form. Here \partial, \bar\partial are the Dolbeault operators. The function \rho is called a Kähler potential for \omega. Conversely, by the complex version of the
Poincaré lemma In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed ''p''-form on an open ball in R''n'' is exact for ''p'' ...
, known as the local \partial \bar \partial-lemma, every Kähler metric can locally be described in this way. That is, if (X,\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 \rho on U such that _U=(i/2)\partial\bar\partial \rho. Here \rho is called a local Kähler potential for \omega. 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 In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
class. This is a consequence of the \partial \bar \partial-lemma from
Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
. Namely, if (X,\omega) is a compact Kähler manifold, then the cohomology class
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
in H_^2(X) is called a Kähler class. Any other representative of this class, \omega' say, differs from \omega by \omega' = \omega + d\beta for some one-form \beta. The \partial \bar \partial-lemma further states that this exact form d\beta may be written as d\beta = i \partial \bar \partial \varphi for a smooth function \varphi: X\to \mathbb. In the local discussion above, one takes the local Kähler class
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
0 on an open subset U\subset X, and by the Poincaré lemma any Kähler form will locally be cohomologous to zero. Thus the local Kähler potential \rho is the same \varphi for
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
0 locally. In general if
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
/math> is a Kähler class, then any other Kähler metric can be written as \omega_\varphi = \omega + i \partial \bar \partial \varphi for such a smooth function. This form is not automatically a positive form, so the space of Kähler potentials for the class
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
/math> is defined as those positive cases, and is commonly denoted by \mathcal: :\mathcal_ := \. If two Kähler potentials differ by a constant, then they define the same Kähler metric, so the space of Kähler metrics in the class
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
/math> can be identified with the quotient \mathcal/\mathbb. The space of Kähler potentials is a
contractible space In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within t ...
. 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

For a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
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 :\mathrm(Y)=\frac\int_Y \omega^r, 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.


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 \partial, \bar \partial and their adjoints, the Laplacians \Delta_d, \Delta_, \Delta_, and the ''Lefschetz operator'' L := \omega \wedge - and its adjoint, the ''contraction operator'' \Lambda = L^*. 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 n, the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
on smooth r-forms is defined by \Delta_d=dd^*+d^*d where d is the exterior derivative and d^*=-(-1)^\star d\, \star, where \star is the
Hodge star operator In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a Dimension (vector space), finite-dimensional orientation (vector space), oriented vector space endowed with a Degenerate bilinear form, nonde ...
. (Equivalently, d^* is the adjoint of d with respect to the ''L''2 inner product on r-forms with compact support.) For a Hermitian manifold X, d and d^* are decomposed as :d=\partial+\bar,\ \ \ \ d^*=\partial^*+\bar^*, and two other Laplacians are defined: :\Delta_=\bar\bar^*+\bar^*\bar,\ \ \ \ \Delta_\partial=\partial\partial^*+\partial^*\partial. If X is Kähler, the Kähler identities imply these Laplacians are all the same up to a constant: :\Delta_d=2\Delta_=2\Delta_\partial . These identities imply that on a Kähler manifold X, :\mathcal H^r(X)=\bigoplus_\mathcal H^(X), where \mathcal H^r is the space of harmonic r-forms on X (forms \alpha with \Delta\alpha=0) and \mathcal H^ is the space of harmonic (p,q)-forms. That is, a differential form \alpha is harmonic if and only if each of its (p,q)-components is harmonic. Further, for a compact Kähler manifold X,
Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
gives an interpretation of the splitting above which does not depend on the choice of Kähler metric. Namely, the
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
H^r(X,\mathbf) of X with complex coefficients splits as a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of certain coherent sheaf cohomology groups: :H^r(X,\mathbf)\cong\bigoplus_H^q(X,\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^(X) be the complex vector space H^q(X,\Omega^p), which can be identified with the space \mathcal H^(X) of harmonic forms with respect to a given Kähler metric. The Hodge numbers of X are defined by h^(X)=\mathrm_H^(X). The Hodge decomposition implies a decomposition of the
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s of a compact Kähler manifold X in terms of its Hodge numbers: :b_r=\sum_h^. The Hodge numbers of a compact Kähler manifold satisfy several identities. The Hodge symmetry h^=h^ holds because the Laplacian \Delta_d is a real operator, and so H^=\overline. The identity h^=h^ can be proved using that the Hodge star operator gives an isomorphism H^\cong \overline. It also follows from
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Ale ...
.


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. A related result is that every compact Kähler manifold is
formal Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements ( forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal atti ...
in the sense of rational homotopy theory. The question of which groups can be
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
s 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
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s 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 In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
arises as the fundamental group of some compact complex manifold of dimension 3. (Conversely, the fundamental group of any
closed manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The onl ...
is finitely presented.)


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 In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...
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. Many properties of Kähler manifolds hold in the slightly greater generality of \partial \bar \partial-manifolds, that is compact complex manifolds for which the \partial \bar \partial-lemma holds. In particular the Bott–Chern cohomology is an alternative to the
Dolbeault cohomology In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohom ...
of a compact complex manifolds, and they are isomorphic if and only if the manifold satisfies the \partial \bar \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. 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 In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
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.


Kähler–Einstein manifolds

A Kähler manifold is called Kähler–Einstein if it has constant
Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
. Equivalently, the Ricci curvature tensor is equal to a constant λ times the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, Ric = ''λg''. The reference to Einstein comes from
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, which asserts in the absence of mass that spacetime is a 4-dimensional
Lorentzian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...
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 A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
) in . It follows that a compact Kähler–Einstein manifold ''X'' must have
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle \Omega on V. Over the complex numbers, it is ...
''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 Fano () is a city and ''comune'' of the province of Pesaro and Urbino in the Marche region of Italy. It is a beach resort southeast of Pesaro, located where the ''Via Flaminia'' reaches the Adriatic Sea. It is the third city in the region by pop ...
, 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 Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...
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 Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth function, smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähl ...
, and Song Sun proved the Yau–
Tian Tian () is one of the oldest Chinese terms for heaven and a key concept in Chinese mythology, philosophy, and cosmology. During the Shang dynasty (17th―11th century BCE), the Chinese referred to their highest god as '' Shangdi'' or ''Di'' (, ...
–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.


Holomorphic sectional curvature

The deviation of a Riemannian manifold ''X'' from the standard metric on Euclidean space is measured by
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...
, 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 A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...
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,\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 \mathbb\to X is constant). If ''X'' happens to be compact, then this is equivalent to the manifold being Kobayashi hyperbolic. On the other hand, if (X,\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. (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.


Examples

# 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
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
in a canonical way; this is known as the existence of
isothermal coordinates In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric ...
. 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 In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
CP''n'', the Fubini–Study metric. One description involves the
unitary group Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semi ...
, 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 space In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian ...
s of compact type, such as
Grassmannian In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...
s. 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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
(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 space In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian ...
s 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 In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field ...
is Kähler (by Siu).


See also

*
Almost complex manifold In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...
* Hyperkähler manifold * Quaternion-Kähler manifold * K-energy functional


Notes


References

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External links

* * {{DEFAULTSORT:Kahler manifold Riemannian manifolds Algebraic geometry Complex manifolds Symplectic geometry