- EXAMPLES OF KAHLER-RICCI SOLITONS 99
]pm-l = (II~=l cpa(U:i<))/ ~,where, for example,
cp1 ( U1 ) 3 ( z1, ... , Zn-1 ) ~ ( -,^1 -, Z2 ... , --Zn-1 ) E cp2(U2).
z1 z1 z1
Given k E N, we formally identify n»^1 = e U { oo} and define the k-twisted
bundle
Ff: = (II~=l (Ua X n»^1 )) / '"'"',
where Ua X TID^1 3 ([x1, ... ,xn];~)'"'"' ([y1, ... ,yn];17) E U13 x n»^1 if and only if
[x1, ... , xn] = [Y1, ... , Yn] and 17 = (x"' Ya )k~ for each a. Equivalently, one may
definewhere, for example,tp1 ( U1 ) X JPl^1 3 ( z1, ... , Zn-1; ( ) ~ ( -, -,^1 z2 ... , --; Zn-1 Z1 k () E cp2(U2 ) X JPl^1 •
z1 z1 z1
Notice that So= {[x1, ... , Xn]; O} and 800 = {[x1, ... , Xn]; oo} are two global
sections of :Ff:.
The key to constructing Kahler-Ricci solitons on J="'f: (as well as examples
on other topologies to be considered below) will be to find a Kahler potential
on en\{O} satisfying certain symmetries and boundary conditions. To see
why this is so, let F'f: = :F'f:\(So u Boo) and define 'ljJ: en\{O}-+ F'f: so that'ljJ : (x1, ... , Xn) 1--t ([x1, ... , Xn]; x~)
if Xa #-0. It is easy to see that ([x1, ... , xn]; x~) '"'"' ([x1, ... , Xn]; x~) when-
ever Xa #- 0 and x13 #-0, hence that 'ljJ is well defined. The map 'ljJ is clearly
surjective. If 'lj;(x1, ... , xn) = 'lj.J(y1, ... , Yn), where, say Xa #- 0 and Y/3 #-0,
then
(Xl , ... , Xa-1, Xa+i, ... , Xn;x~) ~ (Yl , ... , Y/3-1, Y/3+1, ... , Yn;y~).
Xa Xa Xa Xa Y/3 Y/3 Y/3 Y/3
The equivalence relation ~ then implies that y~ = x~, hence that Y/3 = Bx13
for some k-th root of unity e. Because [x1, ... , xn] = [y1, ... , Yn], it follows
that y 7 = Bx 7 for all 1 = 1, ... , n, hence that 'ljJ is a k-to-one map. Therefore,
a Kahler potential Pon en\ {O} will induce a well-defined Kahler metric on
F'f: provided that 88P(Bx1, ... , Bxn) = 8DP(x1, ... , Xn)·
With these considerations in mind, our method will be to construct a
suitable Kahler potential P : en\ {O} -+ IR whose asymptotics as lzl -+ 0
and lzl -+ oo ensure that the induced metric extends smoothly to So and
800 • If we are interested in shrinking solitons, what properties should P
possess? Let's suppose that P determines a Kahler metric g. As above,
we take the Kahler and Ricci forms to be w = A9a(3 dz°' /\ dz!3 and p =
ARa/3 dz°' /\ dz/3, respectively. Then (locally) we have
()2
g--a/3 - oz°'oz/3 p