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 .
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]).
|[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|
Kähler-Einstein metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%A4hler-Einstein_metric&oldid=11694