Kähler-Einstein metric

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