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A [[Kähler metric|Kähler metric]] on a [[Complex manifold|complex manifold]] (or orbifold) whose [[Ricci tensor|Ricci tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k1200301.png" /> is proportional to the [[Metric tensor|metric tensor]]:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k1200302.png" /></td> </tr></table>
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This proportionality is an analogue of the Einstein field equation in general relativity. The following conjecture is due to E. Calabi: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k1200303.png" /> be a compact connected complex manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k1200304.png" /> its first [[Chern class|Chern class]]; then
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A [[Kähler metric|Kähler metric]] on a [[Complex manifold|complex manifold]] (or orbifold) whose [[Ricci tensor|Ricci tensor]] $\operatorname { Ric } ( \omega )$ is proportional to the [[Metric tensor|metric tensor]]:
  
a) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k1200305.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k1200306.png" /> carries a unique (Ricci-negative) Kähler–Einstein metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k1200307.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k1200308.png" />;
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\begin{equation*} \operatorname { Ric } ( \omega ) = \lambda \omega. \end{equation*}
  
b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k1200309.png" />, then any Kähler class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003010.png" /> admits a unique (Ricci-flat) Kähler–Einstein metric such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003011.png" />.
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This proportionality is an analogue of the Einstein field equation in general relativity. The following conjecture is due to E. Calabi: Let $M$ be a compact connected complex manifold and $c _ { 1 } ( M ) _ { \mathbf{R} }$ its first [[Chern class|Chern class]]; then
  
This conjecture was solved affirmatively by T. Aubin [[#References|[a1]]] and S.T. Yau [[#References|[a8]]] via studies of complex Monge–Ampère equations, and Kähler–Einstein metrics play a very important role not only in [[Differential geometry|differential geometry]] but also in [[Algebraic geometry|algebraic geometry]]. The affirmative solution of this conjecture gives, for instance, the Bogomolov decomposition for compact Kähler manifolds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003012.png" />. It also implies (see [[#References|[a2]]], [[#References|[a3]]]):
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a) if $c _ { 1 } ( M ) _ { \mathbf{R} } < 0$, then $M$ carries a unique (Ricci-negative) Kähler–Einstein metric $\omega$ such that $\operatorname { Ric } ( \omega ) = - \omega$;
  
1) Any Kähler manifold homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003013.png" /> is biholomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003014.png" />. Any compact complex surface homotopically equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003015.png" /> is biholomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003016.png" />.
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b) if $c _ { 1 } ( M ) _ { \mathbf{R} } = 0$, then any Kähler class of $M$ admits a unique (Ricci-flat) Kähler–Einstein metric such that $\operatorname { Ric } ( \omega ) = 0$.
  
2) In the Miyaoka–Yau inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003017.png" />, for a compact complex surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003018.png" /> of general type, equality holds if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003019.png" /> is covered by a ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003020.png" />.
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This conjecture was solved affirmatively by T. Aubin [[#References|[a1]]] and S.T. Yau [[#References|[a8]]] via studies of complex Monge–Ampère equations, and Kähler–Einstein metrics play a very important role not only in [[Differential geometry|differential geometry]] but also in [[Algebraic geometry|algebraic geometry]]. The affirmative solution of this conjecture gives, for instance, the Bogomolov decomposition for compact Kähler manifolds with $c _ { 1 } ( M ) _ { \mathbf{R} } = 0$. It also implies (see [[#References|[a2]]], [[#References|[a3]]]):
  
For a Fano manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003021.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003022.png" /> is a compact complex manifold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003023.png" />), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003024.png" /> be the identity component of the group of all holomorphic automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003025.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003026.png" /> be the set of all Kähler–Einstein metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003029.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003031.png" /> consists of a single <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003032.png" />-orbit (see [[#References|[a5]]]). Moreover, the following obstructions to the existence of Kähler–Einstein metrics are known (cf. [[#References|[a5]]], [[#References|[a6]]]):
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1) Any Kähler manifold homeomorphic to $\mathbf{CP} ^ { n }$ is biholomorphic to $\mathbf{CP} ^ { n }$. Any compact complex surface homotopically equivalent to $\mathbf{CP} ^ { 2 }$ is biholomorphic to $\mathbf{CP} ^ { 2 }$.
  
Matsushima's obstruction. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003034.png" /> is a reductive algebraic group (cf. also [[Reductive group|Reductive group]]).
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2) In the Miyaoka–Yau inequality $c _ { 1 } ( S ) ^ { 2 } \leq 3 c_ {  2 } ( S )$, for a compact complex surface $S$ of general type, equality holds if and only if $S$ is covered by a ball in $\mathbf{C} ^ { 2 }$.
  
Futaki's obstruction. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003035.png" />, then Futaki's character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003036.png" /> is trivial.
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For a Fano manifold $M$ (i.e., $M$ is a compact complex manifold with $c _ { 1 } ( M ) _ { \mathbf{R} } > 0$), let $G$ be the identity component of the group of all holomorphic automorphisms of $M$. Let $\cal E$ be the set of all Kähler–Einstein metrics $\omega$ on $M$ such that $\operatorname { Ric } ( \omega ) = \omega$. If $\mathcal{E} \neq \emptyset$, then $\cal E$ consists of a single $G$-orbit (see [[#References|[a5]]]). Moreover, the following obstructions to the existence of Kähler–Einstein metrics are known (cf. [[#References|[a5]]], [[#References|[a6]]]):
  
Recently (1997), G. Tian [[#References|[a7]]] showed some relationship between the existence of Kähler–Einstein metrics on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003037.png" /> and stability of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003038.png" />, and gave an example of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003039.png" /> with no non-zero holomorphic vector fields satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003040.png" />.
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Matsushima's obstruction. If $\mathcal{E} \neq \emptyset$, then $G$ is a reductive algebraic group (cf. also [[Reductive group|Reductive group]]).
  
The Poincaré metric on the unit open disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003041.png" /> (cf. [[Poincaré model|Poincaré model]]) and the [[Fubini–Study metric|Fubini–Study metric]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120030/k12003042.png" /> are both typical examples of Kähler–Einstein metrics. For more examples, see [[Kähler–Einstein manifold|Kähler–Einstein manifold]].
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Futaki's obstruction. If $\mathcal{E} \neq \emptyset$, then Futaki's character $F _ { M } : G \rightarrow \mathbf{C} ^ { * }$ is trivial.
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 +
Recently (1997), G. Tian [[#References|[a7]]] showed some relationship between the existence of Kähler–Einstein metrics on $M$ and stability of the manifold $M$, and gave an example of an $M$ with no non-zero holomorphic vector fields satisfying $\mathcal{E} = \emptyset$.
 +
 
 +
The Poincaré metric on the unit open disc $\{ z \in \mathbf{C} : | z | < 1 \}$ (cf. [[Poincaré model|Poincaré model]]) and the [[Fubini–Study metric|Fubini–Study metric]] on $\mathbf{CP} ^ { n }$ are both typical examples of Kähler–Einstein metrics. For more examples, see [[Kähler–Einstein manifold|Kähler–Einstein manifold]].
  
 
For the relationship between Kähler–Einstein metrics and multiplier ideal sheaves, see [[#References|[a4]]]. See, for instance, [[#References|[a2]]] for moduli spaces of Kähler–Einstein metrics. Finally, Kähler metrics of constant scalar curvature and extremal Kähler metrics are nice generalized concepts of Kähler–Einstein metrics (cf. [[#References|[a2]]]).
 
For the relationship between Kähler–Einstein metrics and multiplier ideal sheaves, see [[#References|[a4]]]. See, for instance, [[#References|[a2]]] for moduli spaces of Kähler–Einstein metrics. Finally, Kähler metrics of constant scalar curvature and extremal Kähler metrics are nice generalized concepts of Kähler–Einstein metrics (cf. [[#References|[a2]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T. Aubin,   "Nonlinear analysis on manifolds" , Springer (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.L. Besse,   "Einstein manifolds" , Springer (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.P. Bourguignon,   et al.,   "Preuve de la conjecture de Calabi" ''Astérisque'' , '''58''' (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A.M. Nadel,   "Multiplier ideal sheaves and existence of Kähler–Einstein metrics of positive scalar curvature" ''Ann. of Math.'' , '''132''' (1990) pp. 549–596</TD></TR><TR><TD valign="top">[a5]</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">[a6]</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">[a7]</TD> <TD valign="top"> G. Tian,   "Kähler–Einstein metrics with positive scalar curvature" ''Invent. Math.'' , '''137''' (1997) pp. 1–37</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> S.-T. Yau,   "On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation I" ''Commun. Pure Appl. Math.'' , '''31''' (1978) pp. 339–411</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> T. Aubin, "Nonlinear analysis on manifolds" , Springer (1982)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> A.L. Besse, "Einstein manifolds" , Springer (1987) {{MR|0867684}} {{ZBL|0613.53001}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> J.P. Bourguignon, et al., "Preuve de la conjecture de Calabi" ''Astérisque'' , '''58''' (1978)</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> A.M. Nadel, "Multiplier ideal sheaves and existence of Kähler–Einstein metrics of positive scalar curvature" ''Ann. of Math.'' , '''132''' (1990) pp. 549–596</td></tr><tr><td valign="top">[a5]</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">[a6]</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">[a7]</td> <td valign="top"> G. Tian, "Kähler–Einstein metrics with positive scalar curvature" ''Invent. Math.'' , '''137''' (1997) pp. 1–37</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> S.-T. Yau, "On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation I" ''Commun. Pure Appl. Math.'' , '''31''' (1978) pp. 339–411</td></tr></table>

Latest revision as of 16:09, 27 January 2024

A Kähler metric on a complex manifold (or orbifold) whose Ricci tensor $\operatorname { Ric } ( \omega )$ is proportional to the metric tensor:

\begin{equation*} \operatorname { Ric } ( \omega ) = \lambda \omega. \end{equation*}

This proportionality is an analogue of the Einstein field equation in general relativity. The following conjecture is due to E. Calabi: Let $M$ be a compact connected complex manifold and $c _ { 1 } ( M ) _ { \mathbf{R} }$ its first Chern class; then

a) if $c _ { 1 } ( M ) _ { \mathbf{R} } < 0$, then $M$ carries a unique (Ricci-negative) Kähler–Einstein metric $\omega$ such that $\operatorname { Ric } ( \omega ) = - \omega$;

b) if $c _ { 1 } ( M ) _ { \mathbf{R} } = 0$, then any Kähler class of $M$ admits a unique (Ricci-flat) Kähler–Einstein metric such that $\operatorname { Ric } ( \omega ) = 0$.

This conjecture was solved affirmatively by T. Aubin [a1] and S.T. Yau [a8] via studies of complex Monge–Ampère equations, and Kähler–Einstein metrics play a very important role not only in differential geometry but also in algebraic geometry. The affirmative solution of this conjecture gives, for instance, the Bogomolov decomposition for compact Kähler manifolds with $c _ { 1 } ( M ) _ { \mathbf{R} } = 0$. It also implies (see [a2], [a3]):

1) Any Kähler manifold homeomorphic to $\mathbf{CP} ^ { n }$ is biholomorphic to $\mathbf{CP} ^ { n }$. Any compact complex surface homotopically equivalent to $\mathbf{CP} ^ { 2 }$ is biholomorphic to $\mathbf{CP} ^ { 2 }$.

2) In the Miyaoka–Yau inequality $c _ { 1 } ( S ) ^ { 2 } \leq 3 c_ { 2 } ( S )$, for a compact complex surface $S$ of general type, equality holds if and only if $S$ is covered by a ball in $\mathbf{C} ^ { 2 }$.

For a Fano manifold $M$ (i.e., $M$ is a compact complex manifold with $c _ { 1 } ( M ) _ { \mathbf{R} } > 0$), let $G$ be the identity component of the group of all holomorphic automorphisms of $M$. Let $\cal E$ be the set of all Kähler–Einstein metrics $\omega$ on $M$ such that $\operatorname { Ric } ( \omega ) = \omega$. If $\mathcal{E} \neq \emptyset$, then $\cal E$ consists of a single $G$-orbit (see [a5]). Moreover, the following obstructions to the existence of Kähler–Einstein metrics are known (cf. [a5], [a6]):

Matsushima's obstruction. If $\mathcal{E} \neq \emptyset$, then $G$ is a reductive algebraic group (cf. also Reductive group).

Futaki's obstruction. If $\mathcal{E} \neq \emptyset$, then Futaki's character $F _ { M } : G \rightarrow \mathbf{C} ^ { * }$ is trivial.

Recently (1997), G. Tian [a7] showed some relationship between the existence of Kähler–Einstein metrics on $M$ and stability of the manifold $M$, and gave an example of an $M$ with no non-zero holomorphic vector fields satisfying $\mathcal{E} = \emptyset$.

The Poincaré metric on the unit open disc $\{ z \in \mathbf{C} : | z | < 1 \}$ (cf. Poincaré model) and the Fubini–Study metric on $\mathbf{CP} ^ { n }$ are both typical examples of Kähler–Einstein metrics. For more examples, see Kähler–Einstein manifold.

For the relationship between Kähler–Einstein metrics and multiplier ideal sheaves, see [a4]. See, for instance, [a2] for moduli spaces of Kähler–Einstein metrics. Finally, Kähler metrics of constant scalar curvature and extremal Kähler metrics are nice generalized concepts of Kähler–Einstein metrics (cf. [a2]).

References

[a1] T. Aubin, "Nonlinear analysis on manifolds" , Springer (1982)
[a2] A.L. Besse, "Einstein manifolds" , Springer (1987) MR0867684 Zbl 0613.53001
[a3] J.P. Bourguignon, et al., "Preuve de la conjecture de Calabi" Astérisque , 58 (1978)
[a4] A.M. Nadel, "Multiplier ideal sheaves and existence of Kähler–Einstein metrics of positive scalar curvature" Ann. of Math. , 132 (1990) pp. 549–596
[a5] T. Ochiai, et al., "Kähler metrics and moduli spaces" , Adv. Stud. Pure Math. , 18–II , Kinokuniya (1990)
[a6] Y.-T. Siu, "Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics" , Birkhäuser (1987)
[a7] G. Tian, "Kähler–Einstein metrics with positive scalar curvature" Invent. Math. , 137 (1997) pp. 1–37
[a8] S.-T. Yau, "On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation I" Commun. Pure Appl. Math. , 31 (1978) pp. 339–411
How to Cite This Entry:
Kähler-Einstein metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%A4hler-Einstein_metric&oldid=23346
This article was adapted from an original article by Toshiki Mabuchi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article