- EXISTENCE OF KAHLER-EINSTEIN METRICS 73
canonical case.) By scaling the metric, we may assume [w] = ci(M). ByLemma 2.26 there exists a smooth function f : M -+ IR such that
p - 27rw = Po8f.
(One can make f unique by the normalization JM e-f dμ = 1.) Let TJ(M)denote the space of (real) holomorphic vector fields on M. The Futaki
functional F[w] : TJ(M) -+ <C is defined byF[w] (V) ~ JM V (f) dμ = JM (V, \7 f) dμ.Futaki [147] showed that F[w] is well defined, i.e., that it depends only on
the homology class [w]. It is then clear that if M admits a Kahler-Einstein
metric, then F[w] vanishes. However, Tian has shown that TJ(M) = 0 (which
implies F[w] = 0) does not imply there exists a Kahler-Einstein metric.
The following uniqueness result in the ci > 0 case was proved by Bando
and Mabuchi [21].
THEOREM 2.32 (Uniqueness of Kahler-Einstein metrics). Let (Mn, g)be a closed Kahler manifold with c1 (M) > 0. The Kahler-Einstein metric
(with positive scalar curvature), if it exists, is unique up to scaling and the
pull back by a biholomorphism of M.
The following result was proved by Andreotti and Frankel [144] for n =
2, Mabuchi [260] for n = 3, and Mori [271] and Siu and Yau [336] in all
dimensions; Mori proved a more general algebraic-geometric result. Work
on characterizing cpn was done by Kobayashi and Ochiai [237].
THEOREM 2.33 (Frankel Conjecture). If (Mn, g) is a closed Kahler man-ifold with positive bisectional curvature, then Mn is biholomorphic to cpn.
In fact, if the bisectional curvature is nonnegative everywhere and posi-
tive at some point, then Mis biholomorphic to <CPn.
When n = 2 and c1 (M) > 0, we have the following (see Tian [345]).
THEOREM 2.34 (ci > 0 surfaces: R > 0 Kahler-Einstein metrics). If
M^2 is a closed complex surface with c1 (M) > 0 and the Lie algebra of the
automorphism group is reductive, then there exists a Kahler-Einstein metric
g on M with positive scalar curvature.
REMARK 2.35. Note that such surfaces are biholomorphic to <CP^2 blown
up at p points, where 3 :S p :S 8. On the other hand, for n 2: 2, cpn blown
up at 1 or 2 points does not admit a Kahler-Einstein metric (see p. 156
of Lichnerowicz [254] and Yau [376]). See Section 7 of this chapter for the
existence of Kahler-Ricci solitons on <CP^2 blown up at 1 or 2 points.
There are a number of additional works related to stability and the ex-istence of Kahler-Einstein metrics with c1 > O. Notably Aubin [14], Siu
[335], Nadel [281], Tian [346], Donaldson [129], [130], and Phong and