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A [[Complex manifold|complex manifold]] carrying a [[Kähler–Einstein metric|Kähler–Einstein metric]]. By the uniqueness property of Kähler–Einstein metrics (see [[#References|[a2]]], [[#References|[a4]]]), the concept of a Kähler–Einstein manifold provides a very natural tool in studying the moduli space of compact complex manifolds.
 
A [[Complex manifold|complex manifold]] carrying a [[Kähler–Einstein metric|Kähler–Einstein metric]]. By the uniqueness property of Kähler–Einstein metrics (see [[#References|[a2]]], [[#References|[a4]]]), the concept of a Kähler–Einstein manifold provides a very natural tool in studying the moduli space of compact complex manifolds.
  
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1) Calabi–Yau manifolds. Any compact connected [[Kähler manifold|Kähler manifold]] of complex dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k1200201.png" /> with holonomy in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k1200202.png" /> is called a Calabi–Yau manifold. A Fermat quintic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k1200203.png" /> with a natural Ricci-flat Kähler metric is a typical example of a Calabi–Yau threefold. Interesting subjects, such as mirror symmetry, have been studied for Calabi–Yau threefolds.
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1) Calabi–Yau manifolds. Any compact connected [[Kähler manifold|Kähler manifold]] of complex dimension $n$ with holonomy in $\operatorname{SU} ( n )$ is called a Calabi–Yau manifold. A Fermat quintic in $\mathbf{CP} ^ { 4 }$ with a natural Ricci-flat Kähler metric is a typical example of a Calabi–Yau threefold. Interesting subjects, such as mirror symmetry, have been studied for Calabi–Yau threefolds.
  
2) More generally, Ricci-flat Kähler manifolds are Kähler–Einstein manifolds (cf. also [[Ricci curvature|Ricci curvature]]). For instance, hyper-Kähler manifolds, characterized as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k1200204.png" />-dimensional (possibly non-compact) Kähler manifolds with holonomy in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k1200205.png" />, are Ricci-flat Kähler manifolds (see [[#References|[a1]]], [[#References|[a3]]]). An ALE gravitational instanton, obtained typically as a minimal resolution of an isolated quotient singularity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k1200206.png" />, has the structure of a hyper-Kähler manifold. A K3-surface (cf. [[Surface, K3|Surface, K3]]) is a compact hyper-Kähler manifold.
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2) More generally, Ricci-flat Kähler manifolds are Kähler–Einstein manifolds (cf. also [[Ricci curvature|Ricci curvature]]). For instance, hyper-Kähler manifolds, characterized as $2 m$-dimensional (possibly non-compact) Kähler manifolds with holonomy in $\operatorname{sp} ( m )$, are Ricci-flat Kähler manifolds (see [[#References|[a1]]], [[#References|[a3]]]). An ALE gravitational instanton, obtained typically as a minimal resolution of an isolated quotient singularity in ${\bf C} ^ { 2 } / \Gamma$, has the structure of a hyper-Kähler manifold. A K3-surface (cf. [[Surface, K3|Surface, K3]]) is a compact hyper-Kähler manifold.
  
 
3) Kähler C-spaces. A compact simply connected homogeneous Kähler manifold, called a Kähler C-space, carries a Kähler–Einstein metric with positive scalar curvature and has the structure of a Kähler–Einstein manifold.
 
3) Kähler C-spaces. A compact simply connected homogeneous Kähler manifold, called a Kähler C-space, carries a Kähler–Einstein metric with positive scalar curvature and has the structure of a Kähler–Einstein manifold.
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4) A twistor space of a quaternionic Kähler manifold with positive scalar curvature has the natural structure of a Kähler–Einstein manifold with positive scalar curvature (see [[#References|[a3]]]).
 
4) A twistor space of a quaternionic Kähler manifold with positive scalar curvature has the natural structure of a Kähler–Einstein manifold with positive scalar curvature (see [[#References|[a3]]]).
  
5) Among the almost-homogeneous Kähler manifolds (cf. [[#References|[a1]]]), the hypersurfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k1200207.png" /> and the del Pezzo surfaces (cf. [[#References|[a5]]], [[#References|[a6]]] or [[Cubic hypersurface|Cubic hypersurface]]), there are numerous examples of Kähler–Einstein manifolds with positive scalar curvature.
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5) Among the almost-homogeneous Kähler manifolds (cf. [[#References|[a1]]]), the hypersurfaces in $\mathbf{CP} ^ { n }$ and the del Pezzo surfaces (cf. [[#References|[a5]]], [[#References|[a6]]] or [[Cubic hypersurface|Cubic hypersurface]]), there are numerous examples of Kähler–Einstein manifolds with positive scalar curvature.
  
6) Any complex manifold covered by a bounded homogeneous domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k1200208.png" /> endowed with a Bergman metric (cf. also [[Hyperbolic metric|Hyperbolic metric]]) is a Kähler–Einstein manifold with negative scalar curvature. More generally, a compact complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k1200209.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k12002010.png" /> naturally has the structure of a Kähler–Einstein manifold with negative scalar curvature.
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6) Any complex manifold covered by a bounded homogeneous domain in $\mathbf{C} ^ { n }$ endowed with a Bergman metric (cf. also [[Hyperbolic metric|Hyperbolic metric]]) is a Kähler–Einstein manifold with negative scalar curvature. More generally, a compact complex manifold $M$ with $c _ { 1 } ( M ) _ { \mathbf{R} } &lt; 0$ naturally has the structure of a Kähler–Einstein manifold with negative scalar curvature.
  
 
==Generalization.==
 
==Generalization.==
A compact complex surface with quotient singularities obtained from a minimal [[Algebraic surface|algebraic surface]] of general type by blowing down <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k12002011.png" />-curves has the structure of a Kähler–Einstein orbifold, which is a slight generalization of the notion of a Kähler–Einstein manifold.
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A compact complex surface with quotient singularities obtained from a minimal [[Algebraic surface|algebraic surface]] of general type by blowing down $( - 2 )$-curves has the structure of a Kähler–Einstein orbifold, which is a slight generalization of the notion of a Kähler–Einstein manifold.
  
 
General references for Kähler–Einstein manifolds are [[#References|[a1]]], [[#References|[a2]]] and [[#References|[a4]]].
 
General references for Kähler–Einstein manifolds are [[#References|[a1]]], [[#References|[a2]]] and [[#References|[a4]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.L. Besse, "Einstein manifolds" , Springer (1987) {{MR|0867684}} {{ZBL|0613.53001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> T. Ochiai, et al., "Kähler metrics and moduli spaces" , ''Adv. Stud. Pure Math.'' , '''18–II''' , Kinokuniya (1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S.M. Salamon, "Quaternionic Kähler manifolds" ''Invent. Math.'' , '''67''' (1987) pp. 175–203</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> Y.-T. Siu, "Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics" , Birkhäuser (1987)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G. Tian, "Kähler–Einstein metrics on certain Kähler manifolds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k12002012.png" />" ''Invent. Math.'' , '''89''' (1987) pp. 225–246</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G. Tian, S.-T. Yau, "Kähler–Einstein metrics on complex surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120020/k12002013.png" />" ''Comm. Math. Phys.'' , '''112''' (1987) pp. 175–203</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> A.L. Besse, "Einstein manifolds" , Springer (1987) {{MR|0867684}} {{ZBL|0613.53001}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> T. Ochiai, et al., "Kähler metrics and moduli spaces" , ''Adv. Stud. Pure Math.'' , '''18–II''' , Kinokuniya (1990)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> S.M. Salamon, "Quaternionic Kähler manifolds" ''Invent. Math.'' , '''67''' (1987) pp. 175–203</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> Y.-T. Siu, "Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics" , Birkhäuser (1987)</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> G. Tian, "Kähler–Einstein metrics on certain Kähler manifolds with $C _ { 1 } ( M ) &gt; 0$" ''Invent. Math.'' , '''89''' (1987) pp. 225–246</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> G. Tian, S.-T. Yau, "Kähler–Einstein metrics on complex surfaces with $C _ { 1 } &gt; 0$" ''Comm. Math. Phys.'' , '''112''' (1987) pp. 175–203</td></tr></table>

Latest revision as of 16:55, 1 July 2020

A complex manifold carrying a Kähler–Einstein metric. By the uniqueness property of Kähler–Einstein metrics (see [a2], [a4]), the concept of a Kähler–Einstein manifold provides a very natural tool in studying the moduli space of compact complex manifolds.

Examples.

1) Calabi–Yau manifolds. Any compact connected Kähler manifold of complex dimension $n$ with holonomy in $\operatorname{SU} ( n )$ is called a Calabi–Yau manifold. A Fermat quintic in $\mathbf{CP} ^ { 4 }$ with a natural Ricci-flat Kähler metric is a typical example of a Calabi–Yau threefold. Interesting subjects, such as mirror symmetry, have been studied for Calabi–Yau threefolds.

2) More generally, Ricci-flat Kähler manifolds are Kähler–Einstein manifolds (cf. also Ricci curvature). For instance, hyper-Kähler manifolds, characterized as $2 m$-dimensional (possibly non-compact) Kähler manifolds with holonomy in $\operatorname{sp} ( m )$, are Ricci-flat Kähler manifolds (see [a1], [a3]). An ALE gravitational instanton, obtained typically as a minimal resolution of an isolated quotient singularity in ${\bf C} ^ { 2 } / \Gamma$, has the structure of a hyper-Kähler manifold. A K3-surface (cf. Surface, K3) is a compact hyper-Kähler manifold.

3) Kähler C-spaces. A compact simply connected homogeneous Kähler manifold, called a Kähler C-space, carries a Kähler–Einstein metric with positive scalar curvature and has the structure of a Kähler–Einstein manifold.

4) A twistor space of a quaternionic Kähler manifold with positive scalar curvature has the natural structure of a Kähler–Einstein manifold with positive scalar curvature (see [a3]).

5) Among the almost-homogeneous Kähler manifolds (cf. [a1]), the hypersurfaces in $\mathbf{CP} ^ { n }$ and the del Pezzo surfaces (cf. [a5], [a6] or Cubic hypersurface), there are numerous examples of Kähler–Einstein manifolds with positive scalar curvature.

6) Any complex manifold covered by a bounded homogeneous domain in $\mathbf{C} ^ { n }$ endowed with a Bergman metric (cf. also Hyperbolic metric) is a Kähler–Einstein manifold with negative scalar curvature. More generally, a compact complex manifold $M$ with $c _ { 1 } ( M ) _ { \mathbf{R} } < 0$ naturally has the structure of a Kähler–Einstein manifold with negative scalar curvature.

Generalization.

A compact complex surface with quotient singularities obtained from a minimal algebraic surface of general type by blowing down $( - 2 )$-curves has the structure of a Kähler–Einstein orbifold, which is a slight generalization of the notion of a Kähler–Einstein manifold.

General references for Kähler–Einstein manifolds are [a1], [a2] and [a4].

References

[a1] A.L. Besse, "Einstein manifolds" , Springer (1987) MR0867684 Zbl 0613.53001
[a2] T. Ochiai, et al., "Kähler metrics and moduli spaces" , Adv. Stud. Pure Math. , 18–II , Kinokuniya (1990)
[a3] S.M. Salamon, "Quaternionic Kähler manifolds" Invent. Math. , 67 (1987) pp. 175–203
[a4] Y.-T. Siu, "Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics" , Birkhäuser (1987)
[a5] G. Tian, "Kähler–Einstein metrics on certain Kähler manifolds with $C _ { 1 } ( M ) > 0$" Invent. Math. , 89 (1987) pp. 225–246
[a6] G. Tian, S.-T. Yau, "Kähler–Einstein metrics on complex surfaces with $C _ { 1 } > 0$" Comm. Math. Phys. , 112 (1987) pp. 175–203
How to Cite This Entry:
Kähler-Einstein manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%A4hler-Einstein_manifold&oldid=50081
This article was adapted from an original article by Toshiki Mabuchi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article