98 2. KAHLER-RICCI FLOWare mutually exclusive: applying the Futaki functional F[w] to the holo-
morphic vector field X = grad f, one getsF[wj(X) =JM (X,X) dμ = [[X[[
2> 0.
Moreover, Kahler-Ricci solitons on a compact Kahler manifold (Mn, J)
are unique up to holomorphic automorphisms. (See Tian and Zhu [352,
353, 354].) Specifically, we have the following theorem.THEOREM 2.76 (Uniqueness of Kahler-Ricci solitons). Let (Mn, J) bea compact Kahler manifold. If metrics g and g' on M satisfy (2.95) with
respect to holomorphic vector fields X and X', respectively, then there is
an element t:J in the identity component of the holomorphic automorphism
group such that g = t:J*g' and X = (t:J-^1 )*X'.For recent results on uniqueness and other properties of noncompact-
Kahler-Ricci solitons, see [63, 64], [35] and [78].
One might ask, therefore, whether there exists either a Kahler-Einstein
metric or else a Kahler-Ricci soliton on every compact Kahler manifoldMn with c1 (M) > 0. The answer is yes if n :::; 2. A compact complex
surface with ci > 0 is IfD^2 #klfD^2 for some k E {O, 1, ... , 8}. (Here and below,
Yln = (C]pin is complex projective space.) A Kahler-Einstein metric exists
for k = 0 and 3 :::; k :::; 8. (See Theorem 2.34.) In the remaining cases
k = 1, 2, there is a (non-Einstein) Kahler-Ricci soliton. (See [239], [366],
[47], and Section 7.2 below.) In higher dimensions, however, the answer is
no. There exist 3-dimensional compact complex manifolds that admit no
Kahler-Einstein metric and no holomorphic vector fields, hence no Kahler-
Ricci soliton structure. (See [346, §7] as well as [215, 216] and [278].)
More generally, Tian and Zhu have exhibited a holomorphic invariant that
generalizes the Futaki invariant and acts as an obstruction to the existence of
a Kahler-Ricci soliton metric [354] on a compact complex manifold (Mn, J).
7.2. The Koiso solitons. As was noted in Proposition 1.13 or Propo-
sition A.32, all compact steady or expanding solitons are Einstein. This is
not true for shrinking solitons. The first examples of nontrivial (i.e. non-
Einstein) compact shrinking solitons were discovered by Koiso [239] and
independently by Cao [47]. These are Kahler metrics on certain k-twisted
projective-line bundles IfD^1 '---+ FJ:---* IfDn-l first described by Calabi [44]. We
will discuss their construction in considerable detail, because it serves as a
prototype for later examples.
We begin with Calabi's bundle construction. IfDn-l is covered by n
charts (cpa : Ua --+ cn-^1 ), where Ua = {[x1, ... ,xn] E IfDn-l : Xa =I= O}
an d 'Pa. [. x1, ... ,Xn l f---+ (Xl -, xa ••• ,--,--, Xa-l Xa Xa+l Xa ••• ,-. Xa Xn) (YH vve wn •t e Xa in-•
stead of xa here and in the next paragraph in order to simplify some for-
mulas below.) In particular, one may define complex projective space by