Uniqueness of Kähler-Einstein cone metrics

Publicacions Ma em`a iques, Vol. 44 (2000), 437–448
UNIQUENESS OF K¨
AHLER-EINSTEIN CONE METRICS
Thalia D. Je es
Abs ac
The pu pose o his pape is o desc ibe a me hod o cons uc a
K¨ahle me ic wi h cone singula i y along a di iso and o illus-
a e a ype o maximum p inciple o hese incomple e me ics
by showing ha K¨ahle -Eins ein me ics a e unique in geome ic
H¨olde spaces.
1. In oduc ion
Ou line o esul s. We show ha i Mis a compac complex mani old
o complex dimension wo o g ea e , and Da di iso wi h one i e-
ducible componen and i he cohomology class C1(KM)+αC1(O(D)),
o α∈(0,1), con ains a posi i e ep esen a i e, hen we can cons uc
an ini ial K¨ahle cone me ic ωwi h cone angle α. He e KMdeno es he
canonical bundle o he mani old and O(D) he line bundle associa ed
o he di iso . This me ic is incomple e along he di iso .
Func ions desc ibing geome ic quan i ies will o en be con inuous on
all o M, bu may achie e nonsmoo h ex ema o e he di iso . We de e-
lope a gene alized maximum p inciple o such unc ions. The echnique
is illus a ed by p o ing uniqueness o K¨ahle -Eins ein cone me ics. Fo
he exis ence o such me ics, [JM], mo e special unc ion spaces a e
needed, which simul aneously yield efined egula i y p ope ies a he
di iso , bu o uniqueness i suffices o wo k wi hin he la ge geome -
ic H¨olde spaces. I is use ul o p o ide an example o he echnique in
his mo e gene al and mo e geome ically in ui i e se ing because he
me hod has o he applica ions. In [J], o example, i is used o p o e a
Schwa z Lemma o K¨ahle me ics wi h cone singula i ies.
Backg ound. Singula spaces a e o in e es in diffe en ial geome y,
algeb aic geome y, and in analysis. They occu na u ally in many se -
ings. Fo example, many algeb aic a ie ies a e no smoo h. Diffe en ial
geome e s in e es ed in special me ics on smoo h mani olds will na u-
ally s udy he moduli space o such, and singula i ies o en de elope a
438 T. D. Je es
he bounda y o he moduli space. In e es ing analy ic ea u es a ise in
he esolu ion o geome ic p oblems, and his pape is an example o
such.
Uniqueness o pa ial diffe en ial equa ions, and also he es ima es
needed o p o e exis ence o K¨ahle -Eins ein me ics, o en ely upon
he maximum p inciple. The impo ance o he maximum p inciple is
ha i enables one o deduce, om ellip ic inequali ies, a p io i es i-
ma es on solu ions o diffe en ial equa ions. In he case o a singula o
noncompac space, di ec applica ion o he maximum p inciple may be
impossible. The e is a close ela ionship be ween singula and noncom-
pac spaces, because emo al o he singula se lea es a noncompac
space. One ob ious difficul y in applying he maximum p inciple is ha
a maximum may simply ail o exis . An ins ance o his may be ound
in he pape o Cheng and Yau [CY] whe ein hey p o e exis ence o
K¨ahle -Eins ein me ics on pseudocon ex domains. Indeed, a non i ial
s ep along he way is a gene alized maximum p inciple which asse s,
oughly speaking, he exis ence o a sequence o poin s app oaching he
bounda y o which he fi s de i a i es o he solu ion go o ze o and
he Hessian becomes nega i e semidefini e. In some cases, he p oblem
can be ci cum en ed. Incomple e me ics on he complemen o a di iso
we e s udied by Tian and Yau, [TY], bu es ic ions on he cone angle
make i possible o pass o a fini e b anched co e and apply echniques
simila o he smoo h, compac case.
The cons uc ion desc ibed below yields a me ic wi h singula i y
along a di iso . Because he me ic is incomple e, i is no possible
o ega d he singula se as being ou a infini y; i is eached in fini e
ime. I a unc ion achie es an ex emum o e he di iso , he singula
backg ound me ic allows he ex emum o be achie ed nonsmoo hly o
wi h a cusp shape. In some sense, his phenomenon is he opposi e o
ha encoun e ed by Cheng and Yau; hei unc ion did no achie e a
maximum bu had he co ec shape, while ou s has a maximum bu
wi h he w ong shape. The maximum p inciple is eally an analysis o
he shape o a unc ion, and he echnique desc ibed below consis s in
using a ba ie unc ion o push he maximum off he di iso and in o he
in e io whe e i will be achie ed smoo hly and wi h he co ec shape.
Na u ally, his ba ie unc ion mus be chosen so ha he esul ing
es ima es a e uni o m.
Uniqueness o K¨
ahle -Eins ein Cone Me ics 439
2. Cone me ics
Cone singula i ies a e e y na u al singula i ies and ha e been s udied
om diffe en poin s o iew by many people. In he Riemannian con ex
one migh consul , o example he pape s o Cheege , among hem [C].
The poin o iew he e is o cons uc a singula space by o ming a cone
o e a gi en compac Riemannian mani old, and he au ho desc ibes
he analysis o such a space. The special case o a sphe e has also been
conside ed by T oyano in [T 1], who has also s udied cone singula i ies
om he poin o iew o Riemann su aces and con o mal maps; see
also [T 2].
In his sec ion we desc ibe a cons uc ion o a K¨ahle cone me ic
wi h nega i e cu a u e on a compac complex mani old. An impo an
applica ion o his cons uc ion is o use i as an ini ial app oxima ion
oaK¨ahle -Eins ein cone me ic and hen p o e exis ence by pe u bing
away om i . In join wo k wi h Ra e Mazzeo, [JM], we used his
app oach o p o e he exis ence o K¨ahle -Eins ein me ics wi h cone
singula i ies a leas o ce ain cone angles. To cons uc his cone
me ic, we need o assume some global condi ions, namely, we suppose
ha Mcon ains a smoo h di iso Dwi h one i educible componen ,
and fix a cons an αwi h 0 <α<1. This cons an will be e e ed o
as he cone angle. Now le KMdeno e he canonical bundle o M, and
O(D) he line bundle associa ed o he di iso D. In o de o make su e
ha wha we cons uc is eally a me ic we need o assume ha
C1(KM)+αC1(O(D)) ∈H2
DR(M)
con ains a posi i e defini e eal, closed (1,1) o m. Some choices o M
and D ha sa is y his a e smoo h algeb aic a ie ies Vin CPndesc ibed
as he ze o locus o a homogeneous polynomial o deg ee k>n+ 1, and
D=V∩H, whe e His a hype plane sec ion o CPn ha in e sec s V
in one smoo h i educible componen .
We can now p oceed o cons uc he K¨ahle cone me ic. Le sbe a
defining sec ion o O(D). Make p o isional choices o a smoo h olume
unc ion Von Mand a He mi ian me ic ·in O(D) and w i e down
ˆ
V=V
s2α(1 −s2(1−α))2.
In he denomina o , we only wan he fi s e m o anish, so mul iply
sby a cons an i necessa y so ha s≤δ<1; his is no p oblem
because sis a smoo h sec ion defined on all o he compac mani old M.
We would like o make sense o his as a singula K¨ahle po en ial so ha
440 T. D. Je es
ou cone me ic will be gi en by ω=i∂∂log ˆ
V, so we compu e di ec ly
ωde
=i∂∂ log ˆ
V
=i∂∂log V−iα∂∂log s2−2i∂∂ log(1 −s2(1−α)).
No ing ha a olume unc ion on Mis he same hing as a me ic hin
he an icanonical line bundle K−1
M, and w i ing Θ(K−1
M) o he cu a u e
o his me ic, he fi s e m he e is
i∂∂log V=i∂∂ log h(K−1
M)=−i∂∂ log h(KM)=iΘ(KM).
How can we in e p e he second e m? Locally, suppose ha e0is
a non anishing holomo phic sec ion o O(D), so ha s=s0e0 o a
holomo phic unc ion s0. Then
−iα∂∂log s2=−iα∂∂log |s0|2−iα∂∂log e02.
This is ac ually independen o he local choices, because any o he choi-
ce e1o non anishing sec ion would gi e s=s1e1=(s1g10)e0=s0e0
and
∂∂log |s0|2=∂∂ log |s1|2+∂∂ log |g10|2=∂∂ log |s1|2+0.
This e m gi es a singula cu en suppo ed o e he di iso , since
∂∂log |s0|2=πδds0∧ds0.
We deno e his cu en by TD. So we ha e
ω=i∂∂log ˆ
V=iΘ(KM)+iαΘ(O(D))
−2παTD−2i∂∂ log(1 −s2(1−α)).
Di ec compu a ion shows ha he las e m also p oduces a singula i y;
i looks like s−2α imes a bounded o m.
Now he assump ion ha 2π(C1(KM)) + αC1(O(D))) >0 comes in o
play, because he sum o he fi s wo e ms, iΘ(KM)+iαΘ(O(D)), is a
ep esen a i e o his cohomology class. The e o e, he ini ial choices o
olume unc ion on Mand me ic in O(D) can be made in such a way
ha he sum o he fi s h ee e ms is posi i e defini e. In ac , in local
holomo phic coo dina es (z,w2,... ,w
n) in which D={z=0}, he sum
o he fi s h ee e ms is equi alen o
√−11
|z|2αdz ∧dz +
n

2
dwi∧dwi.
Uniqueness o K¨
ahle -Eins ein Cone Me ics 441
So ωconsis s o a posi i e defini e me ic on Ω de
=M D oge he wi h
a singula e m suppo ed by D.ωis a cu en on all o M, and a
genuine me ic on Ω bu we jus e e o i as a singula me ic on M.
Mo e desc ip i ely, we say i has a cone singula i y and ha αis he cone
angle, because in he zdi ec ion, he su ace wi h me ic (1/|z|2α)dz∧dz
is a cone.
Le us now explain wha i means o say ha such a me ic is K¨ahle -
Eins ein. Since ωdefines a me ic on Ω, one may compu e he Ricci
cu a u e ρo his me ic, and hen ex end as a cu en o all o M.A
local exp ession o ρis
ρ=−√−1∂∂ log de gi,
and again choosing local coo dina es (z,w2,... ,w
n) o which Dappea s
as he ze o se o z, his is
ρ=−i∂∂log |z|−2αb,
whe e bis a smoo h nonze o bounded unc ion which makes sense on all
o M,o
ρ=iα∂∂log |z|2−∂∂ log b=2παTD−i∂∂log b.
Since −ωand ρbo h con ain he singula e m 2παTDsuppo ed by he
di iso , he co ec K¨ahle -Eins ein condi ion is:
ρ=−ω
as cu en s on all o M, and poin wise on Ω.
As in he smoo h case, he exis ence o a K¨ahle -Eins ein cone me ic
may be e o mula ed analy ically as he exis ence o a solu ion u o he
Monge-Amp`e e equa ion. This is he same equa ion as in he smoo h
case, excep in his con ex a K¨ahle -Eins ein me ic is de e mined by a
solu ion uo e he noncompac se Ω.
The de i a ion oo is he same as in appea ance as in he smoo h
case; one has only o emembe ha e e y hing mus be in e p e ed
in he sense o cu en s and dis ibu ions. Excellen e e ences o he
smoo h case a e he expos´e o Bou guignon, [B], and also he lec u e
no es o Siu, [S]. Wi h ω he o iginal K¨ahle cone me ic, we conside
new me ics o he o m
ωde
=ω+i∂∂u
wi h u∈C2,δ
g(Ω), he so-called geome ic H¨olde space, he defini ion
o which will be gi en in he nex sec ion. Fo he momen , suffice i o

442 T. D. Je es
say ha uis a unc ion ha has 2 +δco a ian o geome ic de i a i es
bounded on Ω. Then umus sa is y
de (gi +∂i∂u)
de (gi)=e +u;
we also equi e ω+i∂∂u > 0 so ha he solu ion is a me ic.
3. Geome ic H¨olde spaces
A na u al se ing in which we sol e nonlinea p oblems on Ω is he
geome ic H¨olde spaces Ck,δ
g(Ω). These spaces consis o he unc ions
which a e con inuous on all o Mand whose co a ian de i a i es up o
o de k+δa e bounded on Ω wi h espec o he singula cone me ic ω
cons uc ed abo e. Namely, Ck
g(Ω) consis s in unc ions uwhich a e
con inuous on all o Mand o which
sup
Ω|u|+···+ sup
Ω∇kug
is fini e. No e ha he singula me ic appea s in his exp ession wice
—bo h in he co a ian de i a i e ∇and in he no m ·. Fo he H¨olde
pa we fi s define C0,δ
g(Ω) o consis in hose unc ions u o which
sup
Ω|u|+ sup
p=q∈Ω
|u(z,w2,... ,w
n)−u(z0,w
2,0,... ,w
n,0)|
|z|αδ|z−z0|δ+|w2−w2,0|δ+···+|wn−wn,0|δ
is bounded. He e p=(z,w2,... ,w
n), and q=(z0,w
2,0,... ,w
n,0).
Successi e Ck,δ
g(Ω) a e hen ob ained by eplacing he nume a o by
|X1...X
ku(p)−X1...X
ku(q)|whe e he Xia e ec o fields o which
Xiis bounded on Ω. The defini ions o Ck
g(Ω) and Ck,δ
g(Ω) a e consis-
en wi h each o he because ∇ugis bounded i and only i o e e y
ec o field X he e is a cons an Cso ha
|Xu|≤CXg.
4. Maximum p inciple echnique
To ob ain he uniqueness esul , in he nex sec ion we will use he
maximum p inciple o show ha he diffe ence be ween wo solu ions,
u1−u2, mus be ze o. We demons a e he idea he e by explaining how
o ob ain a C0es ima e. This is also an impo an s ep in he p oo
o exis ence, [JM]. Then in he ollowing sec ion a efinemen gi es
uniqueness.
Uniqueness o K¨
ahle -Eins ein Cone Me ics 443
We begin by ecalling how C0es ima es on solu ions o he Monge-
Amp`e e equa ion o nega i e fi s Che n class we e ob ained in he
smoo h case by Aubin, [A1] and [A2] and by Yau, [Y]. I uis a solu ion
o he Monge-Amp`e e equa ion, hen locally
de (gi +∂i∂u)
de (gi)=e +u.
Remembe ha is de e mined by he o iginal geome y. A a poin P
whe e uachie es a maximum, (∂i∂u) is a nega i e semidefini e He mi -
ian ma ix, and so a his poin ,
e +u(P)=de (gi +∂i∂u)
de (gi)(P)≤1,
and so (P)+u(P)≤0. The e o e, o all x,weha eu(x)≤max{− (x)}.
In ou singula case, i a maximum o uoccu s o e D, i could ha e a
cusp shape, bu because i is an elemen o he unc ion space C2,δ
g(Ω),
we know exac ly how as he de i a i es can blow up. So ou modifica-
ion is o add a unc ion Fwhich jus ails o be in his space, and show
ha uni o m con ol is main ained.
Pu =u+F o an unknown unc ion F o be de e mined. Then
u= −Fso he Monge Amp`e e equa ion becomes
de (gi +∂i∂ −∂i∂F)
de (gi)=e +u.
Suppose achie es a maximum on Ω. Then a ha poin , (∂i∂ )is
nega i e semidefini e, so ha
de (gi +∂i∂ −∂i∂F)
de (gi)≤de (gi −∂i∂F)
de (gi)
o
e +u≤de (gi −∂i∂F)
de (gi),
ha is,
e ≤e− +F·de (gi −∂i∂F)
de (gi).
I he igh hand side can be bounded, hen we will ha e ob ained a
bound o and hence o u. So we can w i e down he condi ions ha
he choice o Fmus sa is y. They a e:
444 T. D. Je es
1. Max occu s on Ω.
2. max u≤max .
3. Fis uni o mly bounded.
4. Fo some C,Cgi ≤∂i∂F, uni o mly.
Le sbe a sec ion wi h s≤1 and pu F=s2β o a posi i e
powe β o be de e mined in a momen . Then =u+Fwill ag ee wi h
ualong D, and s2βwill be a unc ion which is ini ially inc easing in
di ec ions pe pendicula o he di iso . I Finc eases mo e apidly han
any elemen o C2,δ
g(Ω), hen will achie e a maximum on Ω.
We compa e he g adien o F o ha o unc ions in C2,δ
g(Ω); i he
g adien is unbounded wi h espec o he fla cone me ic ω0 hen i
will also be unbounded wi h espec o ωg.
W i e s2β=|z|2βe2βlocally, wi h ea basis sec ion o O(D), o
s2β=|z|2βb. Because eis bounded away om ze o, bis smoo h.
In diagonal coo dina es a one poin
g ad ,g ad g=
∂
∂zi
2
giı.
We compu e he fi s e m i= 1, since i co esponds o he singula
(z,z) di ec ion
∂
∂z(|z|2βb)∂
∂z (|z|2βb)g11
0=β2b2|z|2+4(β−1)+2α+···.
This is unbounded i 2 + 4(β−1) + 2α<0o 2β<1−α. Because F
ises mo e s eeply han ucan all, max u+Foccu s o e Ω. We now
show ha i∂∂F ≥Cωg o some C, no ing fi s he o mula ∂∂e =
e (∂∂ +∂ ∧∂ ). Then
i∂∂s2β=i∂∂eβlog s
=is2β(β∂∂log s2+β2∂log s2∧∂log s2)
≥is2ββ∂∂log s2,
since o a eal- alued o m h,∂h ∧∂h ≥0.
Bu i∂∂log s2=−R(·), so we need o show ha he e exis s C
such ha
−βs2βR(·)≥Cωg
o
βs2βR(·)≤−Cωg.
Since s≤1 and R(·) is bounded, he e is such a cons an C.
Uniqueness o K¨
ahle -Eins ein Cone Me ics 445
I is in e es ing o no e ha Cdepends on he o iginal me ic and on
he cu a u e o he line bundle, while in he smoo h case o cou se i
only depends on he me ic o M. Tha makes sense, because we had
some eedom in how o choose he me ic in he line bundle O(D).
In he abo e ins ance, all ha was needed was some bound o he
maximum and he minimum o u. Tha is no good enough o ob ain a
uniqueness esul , o ha would only show ha he diffe ence be ween
wo solu ions was a mos one. So we need a sha pening o his echnique.
5. Uniqueness o K¨ahle -Eins ein cone me ics
Now u ning o he uniqueness ques ion, he main esul is:
Theo em. Suppose ha u∈C2,δ
g(Ω) is a solu ion o he Monge-Amp`e e
equa ion
(ω+i∂∂u)n=e +uωn,
wi h ω+i∂∂u posi i e defini e, whe e ωis equi alen o
√−11
|z|2αdz ∧dz +
n

2
dwi∧dwi
a he di iso . Then uis unique.
To p o e his, we suppose ha u1and u2a e wo solu ions o he
Monge-Amp`e e equa ion, lying in C2,δ
g(Ω). Tha is, on he in e io Ω,
(ω+i∂∂u1)n=e +u1ωnand
(ω+i∂∂u2)n=e +u2ωn,
wi h bo h solu ion me ics posi i e defini e. I a maximum o minimum
o he diffe ence u2−u1occu s o e he in e io , hen hese can be
handled as in he smoo h case. So we imagine ha hese ex ema occu
o e he di iso . Since his equa ion is nonlinea , u2−u1is no a solu ion,
so he fi s s ep is o find a nonlinea equa ion o which u2−u1is a
solu ion. Simila ly o he expos´e o Bou guignon, [B], equa ing he wo
exp essions o e ωngi es
(ω+i∂∂u1)n=eu1−u2(ω+i∂∂u1+i∂∂(u2−u1))n.
Se ing u=u2−u1and ω1=ω+i∂∂u1, his eads
euωn
1=(ω1+i∂∂u)n.
This is he nonlinea equa ion sol ed by he diffe ence o he wo solu-
ions.