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| In [[mathematics]] and especially [[differential geometry]], a '''Kähler manifold''' is a [[manifold]] with three mutually compatible structures; a [[complex manifold|complex structure]], a [[Riemannian manifold|Riemannian structure]], and a [[symplectic manifold|symplectic structure]].
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| Kähler manifolds find important applications in the field of [[algebraic geometry]] where they represent generalizations of complex [[projective algebraic varieties]] via the [[Kodaira embedding theorem]] {{Harv|Hartshorne|1977}}. They are named after German mathematician [[Erich Kähler]].
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| ==Definitions==
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| Since Kähler manifolds are naturally equipped with several compatible structures, there are many equivalent ways of creating Kähler forms.
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| ===Symplectic viewpoint===
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| A Kähler manifold is a [[symplectic manifold]] <math> (K,\omega) </math> equipped with an [[Almost complex manifold#Integrable almost complex structures|integrable almost-complex structure]] which is [[Almost complex manifold#Compatible triples|compatible]] with the [[symplectic vector space|symplectic form]].<ref name="da Silva">{{Cite book | last1=Canas da Silva | first1=Ana | author1-link=Ana Canas da Silva | title=Lectures on Symplectic Geometry | publisher=Springer | isbn=978-3540421955| year=2008}}</ref>
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| ===Complex viewpoint===
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| A Kähler manifold is a [[Hermitian manifold]] whose associated [[Hermitian manifold#Riemannian_metric_and_associated_form|Hermitian form]] is [[Closed and exact differential forms|closed]]. The closed Hermitian form is called the '''Kähler metric'''.
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| ===Equivalence of definitions===
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| Every Hermitian manifold <math> K </math> is a [[complex manifold]] which comes naturally equipped with a Hermitian form <math> h </math> and an integrable, [[almost complex structure]] <math> J </math>. Assuming that <math> h </math> is closed, there is a [[Hermitian manifold#Riemannian metric and associated form|canonical]] symplectic form defined as <math> \omega = \frac i2 (h - \bar h ) </math> which is compatible with <math> J </math>, hence satisfying the first definition.
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| On the other hand, any symplectic form compatible with an almost complex structure must be a [[complex differential form]] of type <math> (1,1) </math>, written in a coordinate chart <math> (U, z_i) </math> as
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| :<math> \omega = \frac i2 \sum_{j,k} h_{jk} dz_j \wedge d\bar{z_k} </math>
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| for <math> h_{jk} \in C^\infty(U,\mathbb C) </math>. The added assertions that <math> \omega </math> be real-valued, closed, and non-degenerate guarantee that the <math> h_{jk} </math> define Hermitian forms at each point in <math> K </math>.<ref name="da Silva"/>
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| ==Connection between Hermitian and symplectic definitions==
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| Let <math> h </math> be the Hermitian form, <math> \omega </math> the symplectic form, and <math> J </math> the almost complex structure. Since <math> \omega </math> and <math> J </math> are compatible, the new form <math> g(u,v) = \omega(u,Jv) </math> is Riemannian.<ref name="da Silva"/> One may then summarize the connection between these structures via the identity <math>h=g + i\omega</math>.
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| ==Kähler potentials==
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| If <math> K </math> is a complex manifold, it can be shown<ref name="da Silva"/> that every [[Plurisubharmonic function#Oka theorem|strictly plurisubharmonic function]] <math> \rho \in C^\infty(K; \mathbb R)</math> gives rise to a Kähler form as
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| :<math> \omega = \frac i2 \partial \bar\partial \rho </math>
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| where <math> \partial, \bar\partial </math> are the [[Complex differential form#The Dolbeault operators|Dolbeault operators]]. The function <math> \rho </math> is said to be a '''Kähler potential.'''
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| In fact, utilizing the holomorphic version of the [[Closed and exact differential forms#Poincaré lemma|Poincaré lemma]], a partial converse holds true locally. More specifically, if <math> (K,\omega) </math> is a Kähler manifold then about every point <math> p \in K </math> there is a neighbourhood <math> U </math> containing <math> p </math> and a function <math> \rho \in C^\infty(U,\mathbb R) </math> such that <math> \omega\vert_U = i \partial \bar\partial \rho </math> and here <math> \rho </math> is termed a '''(local) Kähler potential.'''
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| ==Ricci tensor and Kähler manifolds==
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| :see [[Ricci tensor#Kähler manifolds|Kähler manifolds in Ricci tensor]].
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| ==The Laplacians on Kähler manifolds==
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| Let <math>\star</math> be the [[Hodge operator]] and then on an differential manifold ''X'' we can define the Laplacian as
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| <math>\Delta_d=dd^*+d^*d</math>
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| where <math>d</math> is the exterior derivative and <math>d^*=-(-1)^{nk}\star d\star</math>. Furthermore if ''X'' is Kähler then <math>d</math> and <math>d^*</math> are decomposed as
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| :<math>d=\partial+\bar{\partial},\ \ \ \ d^*=\partial^*+\bar{\partial}^*</math>
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| and we can define another Laplacians
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| :<math>\Delta_{\bar{\partial}}=\bar{\partial}\bar{\partial}^*+\bar{\partial}^*\bar{\partial},\ \ \ \ \Delta_\partial=\partial\partial^*+\partial^*\partial</math>
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| that satisfy
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| :<math>\Delta_d=2\Delta_{\bar{\partial}}=2\Delta_\partial . </math>
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| From these facts we obtain the [[Hodge decomposition]] (see [[Hodge theory]])
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| :<math>\mathbf{H^r}=\bigoplus_{p+q=r}\mathbf{H}^{p,q}</math>
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| where <math>\mathbf{H^r}</math> is r-degree [[harmonic form]] and <math>\mathbf{H}^{p,q}</math> is {p,q}-degree harmonic form on ''X''. Namely, an differential form <math>\alpha</math> is harmonic if and only if each <math>\alpha^{i,j}</math> belong to the {i,j}-degree harmonic form.
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| Further, if ''X'' is compact then we obtain
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| :<math>H^p(X,\Omega^q)\simeq H^{p,q}_{\bar{\partial}}(X)\simeq\mathbf{H}^{p,q}</math>
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| where <math>H^{p,q}_{\bar{\partial}}(X)</math> is <math>\bar{\partial}</math>-harmonic cohomology group. This means that if <math>\alpha</math> is an differential form with {p,q}-degree there is only one element in {p,q}-harmonic form due to [[Dolbeault cohomology#Dolbeault theorem|Dolbeault theorem]].
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| Let <math>h^{p,q}=\text{dim} H^{p,q}</math>, called Hodge number, then we obtain
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| :<math>b_r=\sum_{p+q=r}h^{p,q},\ \ \ \ h^{p,q}=h^{q,p},\ \ \ \ h^{p,q}=h^{n-p,n-q}.</math>
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| The LHS of the first identity, ''b<sub>r</sub>'', is r-th [[Betti number]], the second identity comes from that since the Laplacian <math>\Delta_d</math> is a real operator <math>H^{p,q}=\overline{H^{q,p}}</math> and the third identity comes from [[Serre duality]].
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| ==Applications==
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| On a Kähler manifold, the associated Kähler form and metric are called '''Kähler–Einstein''' (or sometimes Einstein–Kähler) if its [[Ricci tensor]] is proportional to the [[metric tensor]], <math>R = \lambda g</math>, for some constant λ. This name is a reminder of [[Einstein]]'s considerations about the [[cosmological constant]]. See the article on [[Einstein manifold]]s for more details.
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| Originally the Kähler condition is independent on the Einstein condition, in which Ricci tensor is proportional to Riemannian metric with constant real number. The important point is that if ''X'' is Kähler then [[Christoffel symbol]]s <math>\Gamma^\alpha_{\beta\gamma}</math> vanish and Ricci curvature is much simplified. The Kähler condition, therefore, is closely related with Ricci curvature. In fact Aubin and Yau prove the [[Calabi conjecture]] using the fact that on a compact Kähler manifold with the first [[Chern class]] ''c<sub>1</sub>=0'' there is a unique Ricci-flat Kähler metric in each Kähler class. But in non-compact case the situation turns to be more complicated and the final solution might not be reached.
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| ==Examples==
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| #Complex [[Euclidean space]] '''C'''<sup>''n''</sup> with the standard Hermitian metric is a Kähler manifold.
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| #A torus '''C'''<sup>''n''</sup>/Λ (Λ a full [[lattice (group)|lattice]]) inherits a flat metric from the Euclidean metric on '''C'''<sup>''n''</sup>, and is therefore a [[compact space|compact]] Kähler manifold.
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| #Every Riemannian metric on a [[Riemann surface]] is Kähler, since the condition for ''ω'' to be closed is trivial in 2 (real) dimensions.
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| #[[Complex projective space]] '''CP'''<sup>''n''</sup> admits a homogeneous Kähler metric, the [[Fubini–Study metric]]. An Hermitian form in (the vector space) '''C'''<sup>''n'' + 1</sup> defines a unitary subgroup ''U''(''n'' + 1) in ''GL''(''n'' + 1,''C''); a Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a ''U''(''n'' + 1) action. By elementary linear algebra, any two Fubini–Study metrics are isometric under a projective automorphism of '''CP'''<sup>''n''</sup>, so it is common to speak of "the" Fubini–Study metric.
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| #The induced metric on a [[complex submanifold]] of a Kähler manifold is Kähler. In particular, any [[Stein manifold]] (embedded in '''C'''<sup>''n''</sup>) or projective [[algebraic variety]] (embedded in '''CP'''<sup>''n''</sup>) is of Kähler type. This is fundamental to their analytic theory.
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| #The unit complex ball '''B'''<sup>''n''</sup> admits a Kähler metric called the [[Bergman metric]] which has constant holomorphic sectional curvature.
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| #Every [[K3 surface]] is Kähler (by a theorem of Y.-T. Siu).
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| An important subclass of Kähler manifolds are [[Calabi–Yau manifold]]s.
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| == Properties ==
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| {{Harv|Deligne|Griffiths|Morgan|Sullivan|1975}} showed that all [[Massey product]]s vanish on a Kähler manifold. Manifolds with such vanishing are [[formal manifold|formal]]: their real homotopy type follows ("formally") from their real [[cohomology ring]].
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| ==See also==
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| *[[Hermitian manifold]]
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| *[[Almost complex manifold]]
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| *[[Hyper-Kähler manifold]]
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| *[[Kähler–Einstein metric]]
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| *[[Quaternion-Kähler manifold]]
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| *[[Complex Poisson manifold]]
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| *[[Einstein manifold]]
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| *[[Calabi conjecture]]
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| ==References==
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| <references />
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| *{{citation
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| |doi=10.1007/BF01389853
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| |last1=Deligne|first1=P.|last2=Griffiths|first2=Ph.|last3=Morgan|first3=J.|last4=Sullivan|first4=D., |title=Real homotopy theory of Kähler manifolds|journal=Invent. Math.|volume=29|year=1975|pages=245–274}}
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| *{{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 | id={{MathSciNet | id = 0463157}} | year=1977}}
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| *Alan Huckleberry and Tilman Wurzbacher, eds. ''Infinite Dimensional Kähler Manifolds'' (2001), Birkhauser Verlag, Basel ISBN 3-7643-6602-8.
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| *Andrei Moroianu, ''Lectures on Kähler Geometry'' (2007), London Mathematical Society Student Texts 69, Cambridge ISBN 978-0-521-68897-0.
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| *Andrei Moroianu, ''Lectures on Kähler Geometry'' (2004), http://www.math.polytechnique.fr/~moroianu/tex/kg.pdf
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| *[[André Weil]], ''Introduction à l'étude des variétés kählériennes'' (1958)
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| ==External links==
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| * {{springer|title=Kähler manifold|id=p/k055070}}
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| {{DEFAULTSORT:Kahler manifold}}
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| [[Category:Riemannian manifolds]]
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| [[Category:Algebraic geometry]]
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| [[Category:Complex manifolds]]
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| [[Category:Symplectic geometry]]
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