Kähler-Einstein metric

A Kähler metric on a complex manifold (or orbifold) whose Ricci tensor is proportional to the metric tensor:

This proportionality is an analogue of the Einstein field equation in general relativity. The following conjecture is due to E. Calabi: Let be a compact connected complex manifold and its first Chern class; then

a) if , then carries a unique (Ricci-negative) Kähler–Einstein metric such that ;

b) if , then any Kähler class of admits a unique (Ricci-flat) Kähler–Einstein metric such that .

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 . It also implies (see [a2], [a3]):

1) Any Kähler manifold homeomorphic to is biholomorphic to . Any compact complex surface homotopically equivalent to is biholomorphic to .

2) In the Miyaoka–Yau inequality , for a compact complex surface of general type, equality holds if and only if is covered by a ball in .

For a Fano manifold (i.e., is a compact complex manifold with ), let be the identity component of the group of all holomorphic automorphisms of . Let be the set of all Kähler–Einstein metrics on such that . If , then consists of a single -orbit (see [a5]). Moreover, the following obstructions to the existence of Kähler–Einstein metrics are known (cf. [a5], [a6]):

Matsushima's obstruction. If , then is a reductive algebraic group (cf. also Reductive group).

Futaki's obstruction. If , then Futaki's character is trivial.

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

The Poincaré metric on the unit open disc (cf. Poincaré model) and the Fubini–Study metric on 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)
[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