In this section, we will prove Theorem 1.3. Similar to the previous section, we can reduce the proof to some boundary estimates. Such estimates can be obtained by applying the following extension lemma on the nef case, a corollary of Lemma 6.1 indeed.
Lemma 7.1. LetX be a compact complex manifold, and E=q
i=1 be a simple normal crossing divisor in X. LetH be a positive line bundle over X, and Θ a positive (1,1)-form representedc1(H). Suppose thatL is a line bundle over X such that L|Ei is nef for each i. Then for any ε >0, there exists a smooth metric hε ofL overX such that
−ddcloghε
Ei+εΘ
Ei >0, i= 1, . . . , q.
Proof. Fix an ε >0. Since L|Ei is nef, L|Ei+εH|Ei is positive, for each i. Then the result follows immediately by applying Lemma 6.1 to the line
bundle L+εH.
To apply this lemma, we set
L=KM + [D], H=KM − p
i=1
αi[Di].
Recall that ωK, defined in Section 2, is a positive (1,1)-form representing c1(H). It follows from Lemma 7.1 that for everyε >0, there exists a function
fε∈C∞(M) such that
(7.1)
ωK−
p
i=1
(αi+ 1)ddcloghi+ddcfε
Di >−εωK
Di. Moreover, recall in Section 2 that the background metric is given by
ω =ωK+ 2 p
i=1
σi−1ωDi + 2 p
i=1
σ−i 2dσi∧dcσi,
in which, near D, the last term on the right is dominant in the normal direction. This observation together with (7.1) imply that there exists a positive numberδ=δ(ε) such that
ω− p
i=1
(αi+ 1)ddcloghi+ddcfε>−εω
on Dδ\D, where Dδ =∪pi=1{|si|2 < δ}.
Starting from here, we can prove Theorem 1.3, by the following lemma, which is essentially done by [22, p. 581]. Here we provide more details for the nonemptiness of the continuity method.
Lemma 7.2. Suppose that for each ε >0, there exists a constant δ= δ(ε)>0 and a bounded functionf∈C2(Dδ) such that
(7.2) ω−
p
i=1
(αi+ 1)ddcloghi+ddcfε>−εω
on Dδ\D. Then there exists an almost-complete K¨ahler–Einstein metric on M.
Let us recall that a K¨ahler–Einstein metricωE onM is said to be almost complete, if there is a sequence{ωk} of complete K¨ahler metrics such that
(i) Ric(ωkn)≤tkωk, with limtk = 1;
(ii) ωk converges toωE inC∞ uniformly on every compact subset ofM.
Proof. Let T◦ =
t∈(1,+∞);(ω+ddcu)n ωn =etu
p
i=1
|si|2aieFhas a solution u∈ U
with u+1 t
p
i=1
ailog|si|2 ∈Ck+2,α(M).
,
where U is the open set given by U =
u∈C2(M); C−1ω < ω+ddcu < Cω for someC >1 . It is sufficient to show that
T◦ = (1,+∞).
In fact, by the Monge–Amp`ere equation, we have for eacht∈T◦, Ric
(ω+ddcu)n
=ω+tddcu
≤t(ω+ddcu).
(7.3)
On the other hand, we can always get the global uniform estimate of e−Cu(n+ Δu),
by the global uniform estimates sup(tu+
ailog|si|2) and infu. Hence, n+ Δu is uniformly bounded on every compact subset of M. Hence, we have a uniform C3-estimate of u on every compact subset. Therefore, for any sequence tk in T◦ with limtk= 1, there is a subsequence of ui that satisfies (7.3) and converges uniformly on every compact subset to a solution u∈C∞(M) of the Monge–Amp`ere equation att= 1. Thus,ω+ddcuis the desired almost-complete K¨ahler–Einstein metric onM.
Hence, it suffices to show thatT◦ is nonempty, and both open and closed in (1,+∞). The openness and closedness can be proved in the same way as in Section 6, except that the global estimates inf(u+ (1/t)
ailog|si|2) and sup(n+ Δu) depend on t because of (7.2). For instance, let us carry out the estimate for inf(u+ (1/t)
ailog|si|2). Let v=tu+
p
i=1
ailog|si|2 ∈Ck+2,α(M).
It suffices to considerv inDδ\D. Note that (7.2) is equivalent to ωu−1
tddc(v−f)>
1−1 +ε t
ω,
whereωu =ω+ddcu. Sincet >1, we can choose an ε >0, say ε= t−1
2 ,
such that 1−(1 +ε)/t >0. Then, by the Monge–Amp`ere equation, e−(v−fε)= ωn
ωuneF+f
≤ 2t
t−1 n
1− 1
ntΔωu(v−fε) n
esup(F+f).
Applying Lemma 3.2 yields that ev−fε≥
t−1 2t
n
e−sup(F+f) on Dδ\D.
Namely,
v≥C16(t)≡nlog t−1
2t
−sup(F +f) + inffε on Dδ\D, whereε= (t−1)/2. Hence,
infv≥min
2 p
i=1
log(δ/2)−supF, C16(t)
.
Similarly, we have
0< n+ Δu < C17(t).
We remark that the positive constantC17(t) blows up whent→1+. Finally, the nonemptiness is settled by the following proposition.
Proposition 7.3. There exists a sufficiently larget∈T◦ so that T◦=∅.
Proof. Let us denote by
ωγ,w =ω+γddcw−γ p
i=1
aiddclog|si|2+ddcF
,
for all γ ∈[0,1] and w∈Ck+2,α(M). Then the Monge–Amp`ere equation given in the definition of T◦ is equivalent to
ωn1/t,v ωn =ev, if we set v=tu+
ailog|si|2+F. Let us fix some large constant C >1.
Let
VC ={(γ, w)∈[0,1]×Ck+2,α(M); C−1ω < ωγ,w < Cω} and define a C∞–map M:VC →Ck,α(M) by
M(γ, w) = logωγ,wn
ωn −w for all (γ, w)∈ VC
Hence, it suffices to show that there exists a pair (γ, v)∈ VC with γ = 0, such that
M(γ, v) = 0.
Note thatM(0,0) = 0. However, the implicit function theorem does not work directly, since the partial Fr´echet derivative ofMwith respect to vis degenerate atγ = 0. We overcome this difficulty by using the Newton’s iter-ation method, which is completely elementary without using the curvature property of ω.
For a small γ = 0, we want to construct a sequence {vl}∞l=0 such that v0 = 0,
(7.4) vl+1 =vl−D2M(γ, vl)−1
M(γ, vl)
, l= 0,1,2, . . . ,
and (γ, vl)∈ VC for all l, whereD2M(γ, v) is the partial Fr´echet derivative of M at (γ, v) with respect tov. Note that
D2M(γ, v) =γΔγ,v−1,
where Δγ,v is the negative Laplacian associated withωγ,v. Hence, for any f ∈Ck,α(M), there exists a unique solution h∈Ck+2,α(M) for
D2M(γ, v)h=f,
as long as γ = 0 and (γ, v)∈ VC. Applying Yau’s maximum principle (Lemma 3.1) and the interior H¨older estimate yields that
(7.5) hk+2,α≤γ−1Ck,αfk,α,
where the constant Ck,α depends only on k, α, dimM and the constant C associated withVC.
On the other hand, by Newton’s method, we have vl+1−vl=−D2M(γ, vl)−1
M(γ, vl)
− M(γ, vl−1)−D2M(γ, vl)(vl−vl−1)
=−D2M(γ, vl)−1· 1
0
(1−τ)D22M(τ vl+ (1−τ)vl−1)(vl−vl−1, vl−vl−1)
dτ.
Here the second-order partial derivative D22M(γ, v)(h, f) =γ2
n i,j,p,q=1
giγ,vq¯ gpγ,v¯j hi¯jfpq¯,
for all h,f ∈Ck+2,α(M) and all (γ, v)∈ VC, and ωγ,v=
√−1 2π
n i,j=1
(gγ,v)i¯jdzi∧d¯zj, n k=1
gγ,vi¯k (gγ,v)j¯k=δij.
Observe that for any (γ, v1), (γ, v2)∈ VC,
(γ, τ v1+ (1−τ)v2)∈ VC for all τ ∈[0,1].
Thus, if (γ, vj)∈ VC for all j < l+ 1, then by (7.5) we obtain vl+1−vlk,α≤ γ
2Ck,αC2
i,j,p,q
giq¯gp¯j(vl−vl−1)i¯j(vl−vl−1)p¯q k,α
≤γCnvl−vl−12k+2,α, (7.6)
where the constantCndepends only onk,α,ωand the constantCassociated withVC.
Since (0,0)∈ VC, there exists a smallδ0>0 such that for anyγ <2δ0, w∈Ck+2,α(M) withγw2,α<2δ0, we have (γ, w)∈ VC. Letv0= 0. Then
M(γ, v0) = logωγ,0n ωn
=−γΔω
p
i=1
ailog|si|2+F
+γ2ϕ,
for someϕ∈ R(M). Let v1 be given by (7.4). In view of (7.5) we have v1k+2,α= −D2M(γ,0)−1(M(γ,0))k+2,α
≤Ck,αC17,
in which the constant C17 is independent of γ. Now we fix a sufficiently small γ such that
0< γ <min
δ0, δ0
2Ck,αC17, 1 2CnCk,αC17
. Then
γv1k+2,α≤min δ0
2, 1 2Cn
;
in particular, (γ, v1)∈ VC. Letv2be given by (7.4). It follows from (7.6) that v2−v1k+2,α≤ 12v1k+2,α,
and
γv2k+2,α≤γv2−v1k+2,α+γv1k+2,α
≤ δ0 4 +δ0
2 = 3 4δ0.
Hence, (γ, v2)∈ VC. By (7.4), we obtain by induction a sequence {vl} sat-isfying that
vl−vl−1k+2,α≤ 1
2l−1v1k+2,α, and
γvlk+2,α≤ l j=1
δ0 2j < δ0,
for all l∈N. In particular, each (γ, vl)∈ VC. The sequence converges in · k+2,α to an elementv∗ inCk+2,α(M), as
∞ j=1
vl−vl−1k+2,α≤∞
j=1
1
2l−1v1k+2,α
≤2v1k+2,α. Furthermore, (γ, v∗)∈ VC since
γv∗k+2,α≤2γv1k+2,α≤δ0. Lettingl→ ∞ in (7.4) yields thatM(γ, v∗) = 0. Hence,
γ−1 ∈T◦ withu1/γ =−γ
ailog|si|2+v∗−F. This completes the proof.
Acknowledgments
First of all, the author would like to thank Professor Shing-Tung Yau. From his lectures and papers the author has learned the upper bound lemma together with other techniques for the Monge–Amp`ere equations. Part of the work was done when the author was at Stanford University; he would like to thank Professor Rick Schoen for his support and encourage. The author would also thank Professors Fangyang Zheng and Matt Stenzel for discussions and their moral support. Finally, he would like to thank the referees for their careful proof reading and helpful questions.
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Department of Mathematics The Ohio State University 1179 University Drive Newark, OH 43055
E-mail address: dwu@math.ohio-state.edu Received May 15, 2007