View source for Kähler-Einstein metric

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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]]:

\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|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 [[#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]]]):

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 [[#References|[a5]]]). Moreover, the following obstructions to the existence of Kähler–Einstein metrics are known (cf. [[#References|[a5]]], [[#References|[a6]]]):

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

====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) {{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>