- EXAMPLES OF KAHLER-RICCI SOLITONS 97
For some other recent work on the Kahler-Ricci fl.ow the reader is re-
ferred to Phong-Sturm [305], [306], Chen [84], Chen and Li [85], Song and
Tian [337], Cascini and La Nave [60], Tian and Zhang [351], and [234].7. Examples of Kahler-Ricci solitons
In this section, we provide a brief and regrettably incomplete sampling
of some results on Kahler-Ricci solitons. These special solutions model
singularities of the Kahler-Ricci fl.ow. A Kahler-Ricci soliton is a Kahler
manifold (Mn,g, J) such that the soliton structure equation(2.95)1
Rc+Ag +2
.cxg = 0
holds for some constant A E IR and some real vector field X which is an in-
finitesimal automorphism (2.1) of the complex structure J. Note that X
is an infinitesimal automorphism if and only if its (1, 0)-part is holomorphic:
0 = V aXf3 = o~°' Xf3. One imposes this requirement for the following reason.
As we saw in Lemma 2.36, a solution of Ricci fl.ow that starts with a Kahler
metric on a complex manifold remains Kahler with respect to the same com-
plex structure. On the other hand, if !.pt is any family of diffeomorphisms of
M, then each pullback r.pt(g) is Kahler with respect to the complex structure
r.pt(J). Now consider the evolving metric h(t) := (1 + At)r.p"tg, where !.pt is
the family of diffeomorphisms generated by 2 (l~.At)X. If Xis an infinitesi-
mal automorphism of the complex structure, then !.fJt ( J) = J, which implies
that h(t) remains Kahler with respect to the same complex structure. Fur-
thermore, using (2.95), it is easy to see that h solves the Kahler-Ricci fl.ow:
gth = -Rc(h).
One may also define a Kahler-Ricci soliton to be a Kahler manifold
(Mn, g, J) together with a constant A E IR and a real vector field X satisfying
the complex soliton equation
1
(2.96) Ra/3 + A9af3 + 2,(£xg)af3 = 0.Equation (2.96) is equivalent to the conjunction of equation (2.95) and the
statement that X is holomorphic. Notice that if we restrict our attention
to gradient solitons (so that X is the gradient of a real-valued function),
then (2.96) is equivalent to (2.95) without any extra hypotheses. (See §2.2
of [142] for the detailed argument.)
7.1. Existence and uniqueness. Any Kahler metric satisfying (2.96)
with X = O is Kahler-Einstein. In this sense, Kahler-Einstein metrics may
be regarded as trivial Kahler-Ricci solitons.^8 So if no Kahler-Einstein met-
ric exists, a natural replacement is a Kahler-Ricci soliton. In fact, existence
of a Kahler-Einstein metric and a nontrivial gradient Kahler-Ricci soliton
(^8) 0£ course, there is nothing 'trivial' about Kahler-Einstein metrics!