Difference between revisions of "Kähler-Einstein metric"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 17:27, 31 March 2012
A Kähler metric on a complex manifold (or orbifold) whose Ricci tensor is proportional to the metric tensor:
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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 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) 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 |
Kähler-Einstein metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%A4hler-Einstein_metric&oldid=24029