Constructions of Kahler-Einstein metrics with negative scalar curvature
CONSTRUCTIONS OF KÄHLER-EINSTEIN METRICS
WITH NEGATIVE SCALAR CURVATURE
Jian Song1 Ben Weinkove2
Johns Hopkins University Harvard University
Department of Mathematics Department of Mathematics
Baltimore, MD 21218 Cambridge, MA 02138
Abstract. We show that on Kähler manifolds with negative first Chern
class, the sequence of algebraic metrics introduced by H. Tsuji converges
uniformly to the Kähler-Einstein metric. For algebraic surfaces of general
type and orbifolds with isolated singularities, we prove a convergence result
for a modified version of Tsuji’s iterative construction.
1. Introduction
AKähler manifold with negative first Chern class admits a unique Kähler-
Einstein metric. This was shown by Yau in his seminal paper on the Calabi
conjecture [Y1], and also independently by Aubin [A]. Yau later posed the
question of whether, in general, Kähler-Einstein metrics could be obtained
as a limit of algebraic metrics induced from embeddings into projective space
[Y2]. Donaldson [D1] showed that if a polarized variety (X,L) with discrete
automorphism group admits a constant scalar curvature Kähler metric, then
there is indeed a sequence of algebraic ‘balanced metrics’ [Zh] converging to
it. Donaldson’s proof makes use of the Tian-Yau-Zelditch expansion of the
Bergman kernel [Ti], [Ze] (see also [C]) and Lu’s [L] computation of the sec-
ond coefficient. Very recently, Donaldson [D2] has described how numerical
approximations to these balanced metrics could be used to compute, to a
good accuracy, explicit Kähler-Einstein metrics on certain varieties.
Tsuji [Ts] has considered a different way of producing Kähler-Einstein
metrics by algebraic approximations. He introduced a new iterative proce-
dure on varieties of general type with the aim of describing (possibly sin-
gular) Kähler-Einstein metrics. In the case when the first Chern class is
The first-named author is on leave for the semester and is visiting MSRI, Berkeley, CA
as a postdoctoral fellow. He is supported in part by National Science Foundation grant
DMS 0604805.
Part of this research was carried out while the second-named author was a short-term
visitor at MSRI in January 2007. He is supported in part by National Science Foundation
grant DMS 0504285.
http://arxiv.org/abs/0704.1005v1
negative, Tsuji proved that his iteration converges, in a certain weak sense,
to the Kähler-Einstein metric. In this paper, we give a uniform convergence
result and describe how his procedure may be modified to obtain results in
the case of algebraic surfaces of general type, and on orbifolds with isolated
singuarities.
We now describe Tsuji’s iteration. Let X be a compact Kähler mani-
fold of complex dimension n with ample canonical bundle KX . Let ωKE =√
(gKE)ijdz
i ∧ dzj be the Kähler-Einstein metric satisfying
Ric(ωKE) = −
∂∂ logωnKE = −ωKE ∈ c1(X).
Fix a Hermitian metric hKE on KX by setting hKE = (det gKE)
Let m0 ≥ 1 be an integer such that Km0X is base point free and let hm0
be any Hermitian metric on Km0X . Define a sequence of Hermitian metrics
hm on K
X for m > m0 inductively as follows. Suppose that hm is a given
Hermitian metric on KmX . To define hm+1, first define an inner product
〈 , 〉Tm+1 on the space of sections of Km+1X by
〈s, t〉Tm+1 =
hm ⊗ s⊗ t, for s, t ∈ H0(X,Km+1X ), (1.1)
where hm⊗s⊗t is regarded as a volume form onX. Now let (σ
m+1, . . . , σ
(Nm+1)
m+1 ),
for Nm+1+1 = dimH
0(X,Km+1X ), be an orthonormal basis of H
0(X,Km+1X )
with respect to this inner product. Then define the Hermitian metric hm+1
on Km+1X by
hm+1 =
(m+ n+ 1)!
(m+ 1)!
m+1 ⊗ σ
Observe that this metric is independent of the choice of orthonormal basis.
We have the following theorem on the convergence of the metrics hm,
strengthening the result given in [Ts].
Theorem 1 Let X be a compact Kähler manifold with ample canonical
bundle. Let hKE, ωKE and the sequence of Hermitian metrics hm be as
above. There exists a constant C depending only on X and hm0 such that
for m ≥ m0,
C logm
m hKE ≤ h1/mm ≤ e
C logm
m hKE. (1.2)
Hence h
m converges uniformly on X to hKE as m→ ∞.
We now describe a modification of Tsuji’s iteration. Let β be a continu-
ous function on a variety X with 0 ≤ β ≤ 1. Here we allow X to have mild
singularities. We also remove the assumption that KX be ample. We only
require that there exists an m0 ≥ 1 such that Km0X is base point free. Let
hm0 be a Hermitian metric on K
X . Given 0 < ε ≤ 1, we inductively define
a sequence of Hermitian metrics hm,ε = hm,ε(β, hm0) on KX as follows. As-
suming that hm,ε is given, define an inner product 〈 , 〉Tm+1,ε on the space
of sections of Km+1X by
〈s, t〉Tm+1,ε =
βεhm,ε ⊗ s⊗ t, for s, t ∈ H0(X,Km+1X ).
Then define the Hermitian metric hm+1,ε on K
hm+1,ε =
(m+ n+ 1)!
(m+ 1)!
m+1,ε ⊗ σ
m+1,ε
where (σ
m+1,ε, . . . , σ
(Nm+1)
m+1,ε ) is an orthonormal basis of H
0(X,Km+1X ) with
respect to the inner product 〈 , 〉Tm+1,ε . We call hm,ε the modified Tsuji
iteration. It depends on ε and β.
First, we consider the case when X is an algebraic surface of general
type. Let E =
iEi be the sum of the nonsingular rational curves Ei of self
intersection −1 (or (−1)-curves, for short) on X. Let τ : X → Xmin be a
holomorphic map blowing down these curves onX, so that Xmin is a minimal
surface of general type. Now if h is any Hermitian metric on KXmin, then
Ω = h−1(z1, . . . , zn)(
−1/2π)ndz1∧dz1∧ · · ·∧dzn∧dzn can be regarded as
a volume form on Xmin. Using coordinates w
i on X, there is a holomorphic
section S−1 of the line bundle [E ] associated to E and vanishing of order one
on E satisfying
(τ∗h)−1 ⊗ |S−1|2
(w1, . . . , wn)
dw1 ∧ dw1 ∧ · · · ∧ dwn ∧ dwn = τ∗Ω,
for any such h.
Xmin has no (−1)-curves, but may have (−2)-curves. Let f : Xmin →
Xcan be its canonical map. Xcan is an surface with ample canonical bundle
KXcan and, at worst, isolated orbifold singularities. By the orbifold version of
the results of [Y1], [A] (see [K], for example), there exists an orbifold Kähler-
Einstein metric ωKE on Xcan, with corresponding Hermitian metric hKE on
KXcan . Define a Kähler metric on Xmin by ωmin = f
∗ωKE and a Hermitian
metric on KXmin by hmin = f
∗hKE. Note that ωmin and hmin are not smooth
in general, although hmin is continuous on Xmin (see [TZ]). Let C =
be the sum of the (−2)-curves on Xmin and let S−2 be a holomorphic section
of the line bundle [C], vanishing of order one on C. Fix a smooth Hermitian
metric hC on [C], and assume that supXmin |S−2|
= 1. Let β be the smooth
function on X defined by β = τ∗|S−2|2hC , and let h∞ = τ
∗hmin.
For this β and some initial Hermitian metric hm0 on K
X , let hm,ε =
hm,ε(β, hm0) be the sequence of Hermitian metrics in the modified Tsuji
iteration as described above. Then we have the following result.
Theorem 2 Let X be an algebraic surface of general type. With hm,ε =
hm,ε(β, hm0) and S−1 as described above, for every sequence εj → 0,
lim sup
h1/mm,εj → h∞ ⊗ |S−1|
−2, as j → ∞,
almost everywhere on X.
Note that since KX = τ
∗KXmin + [E ], we can regard h∞ ⊗ |S−1|−2 as a
singular Hermitian metric on KX .
We now consider the case whenX is an orbifold with isolated singularities
with ample canonical bundle.
Theorem 3 Let (X,ωKE) be a Kähler-Einstein orbifold with KX ample and
only isolated singularities at points p1, . . . , pk. Let β be a continuous function
on X satisfying 0 ≤ β ≤ 1 and β(x) = 0 if and only if x = pi for some i.
Then, with hm,ε = hm,ε(β, hm0) as above, for every sequence εj → 0,
lim sup
h1/mm,εj → hKE, as j → ∞,
almost everywhere on X.
We end the introduction with a couple of remarks.
1. Tsuji’s iteration has some similarities to Donaldson’s TK -iteration (see
[D2], section 2.2.2) which in the case of KX ample should yield a
‘canonically balanced metric’ using sections of a fixed power k of the
canonical line bundle. As k → ∞, the limit of these metrics is expected
to be the Kähler-Einstein metric. On the other hand, Tsuji’s method
is a single iterative process.
2. Donaldson has suggested that the dynamical systems introduced in
[D2] are likely to be discrete approximations to the Kähler-Ricci and
Calabi flows. It would be interesting to know whether Tsuji’s iteration
could be viewed in a similar light.
The outline of the paper is as follows. Our main technique is the peak
section method of [Ti]. This is described in section 2, and extended to
orbifolds with isolated singularities (for related results on the Szegö kernel
for orbifolds, see [S], [DLM], [P]). In sections 3 and 4 we prove the main
theorems.
2. Peak sections
Now suppose that (X,ω) be a compact Kähler manifold of complex di-
mension n with ω ∈ c1(L) for an ample line bundle L. Fix a Hermitian
metric h on L satisfying −
∂∂ log h = ω. Write 〈·, ·〉L2(hm) and ‖ · ‖L2(hm)
for the L2 inner product and norm on H0(X,Lm) with respect to the Her-
mitian metric hm and volume form 1
ωn. We use the following lemma from
[Ti].
Lemma 2.1 There exists m1 > 0 depending only on X, L and h such that
for every x0 ∈ X and m ≥ m1 there is a global holomorphic section sm,x0 of
Lm satisfying the following.
(i) |sm,x0 |2hm(x0) = 1 and
‖sm,x0‖2L2(hm) =
(m+ n)!
(1 + O(m−1)),
where f(m) = O(m−1) means that |f(m)|m ≤ A for a constant A
depending only on X, L and h.
(ii) For t any holomorphic section of Lm which vanishes at x0,
∣〈sm,x0 , t〉L2(hm)
∣ ≤ B
‖sm,x0‖L2(hm)‖t‖L2(hm),
with a constant B depending only on X, L and h.
Proof We outline here the proof of part (i), since we will explicitly refer to
this method in the orbifold case. For (ii) we refer the reader to [Ti]. Pick a
normal coordinate chart (U, (z1, z2, . . . , zn)) for g centered at the point x0.
Let η : [0,∞) → [0, 1] be a cut-off function satisfying η(t) = 1 for t ≤ 1
η(t) = 0 for t ≥ 1 and −4 ≤ η′(t) ≤ 0, |η′′(t)| ≤ 8. Define a weight function
ψ, which for m sufficiently large is supported in U , by setting
ψ(z) = (n+ 2)η
am|z|2
am|z|2
, (2.1)
for z ∈ U , and ψ ≡ 0 outside U , where am = m/(logm)2. A short calculation
shows that √
∂∂ψ(z) ≥ −C(n+ 2)amω, for |z| > 0. (2.2)
Now let ψi be a decreasing sequence of smooth functions on X such that
ψ = lim
ψi and
∂∂ψi ≥ −C(n+ 2)amω, (2.3)
where the constant C may be different from the one in (2.2). It follows that
for sufficiently large m,
∂∂ψi −
∂∂ log hm +Ric(ω) ≥
ω. (2.4)
Choose a local holomorphic section v of L so that |v|2h(x0) = 1 and (∂|v|2h)(x0) =
0. Let w be the smooth local section of Lm defined by
w = ∂
am|z|2
We apply the L2 estimate of Hörmander [H] (cf. Proposition 1.1 of [Ti]) to
the (1,0) form w: there exists a smooth global section u of Lm satisfying
∂u = w and
|u|2hme−ψ
|w|2hme−ψ
Here we are using the fact that the weight function ψ can be approximated
by ψi satisfying (2.4). Observe that w vanishes identically outside the region
2/am ≤ |z|2 ≤ 4/am, and that in U , |w|2hme−ψ ≤ Cam|vm|2hm. It follows that
|u|2hme−ψ
(logm)2
2/am≤|z|2≤4/am
|vm|2hm
From our choice of coordinates, |v|2h(z) = 1− |z|2 +O(|z|3), and so locally
|vm|2hm
)m(√−1
dz1 ∧ dz1 ∧ · · · ∧ dzn ∧ dzn.
Hence
|u|2hm
(logm)2
≤ C(logm)2n−2m− logm/2−n. (2.5)
Now set sm,x0 = η
am|z|2
vm − u. Since
|u|2hme−ψ
< ∞, we have
u(z) = O(|z|2) by the definition of ψ. Hence |sm,x0 |2hm(x0) = 1. Calculate
|sm,x0 |2hm
am|z|2
|vm|2hm
|u|2hm
+ 2Re
am|z|2
vm, u
. (2.6)
From (2.5) the last two terms are O(m−q) for any q. For the first term,
observe that
|z|2≤2/am
|vm|2hm
am|z|2
|vm|2hm
|z|2≤4/am
|vm|2hm
(2.7)
We will make use of the following elementary lemma:
Lemma 2.2 Fix b > 0 and q > 0. Then
|z|2≤b/am
(1−|z|2)mdz1∧dz1∧· · ·∧dzn∧dzn =
(m+ n)!
+O(m−q),
where the term O(m−q) depends only on b, q and n.
Then for any fixed b > 0 and for m sufficiently large,
|z|2≤b/am
|vm|2hm
(m+ n)!
(1 + O(m−1)).
From (2.6) and (2.7) we obtain
|sm,x0 |2hm
(m+ n)!
(1 + O(m−1)),
as required. �
Assume now that (X,ω) is a Kähler orbifold of complex dimension n
with isolated orbifold singularities at points p1, . . . , pk. Then for each x ∈
{p1, . . . , pk} there is an open neighborhood Vx ⊂ X containing x, a finite
subgroup Gx ∈ U(n), a Gx-invariant open neighborhood Ṽx of the origin in
n and a projection map πx : Ṽx → Ṽx/Gx ∼= Vx with πx(0) = x. Let h be an
orbifold Hermitian metric on an orbifold line bundle L with −
∂∂ log h =
ω. We show the following.
Lemma 2.3 There exists m1 > 0 depending only on X, L and h such that
for each m ≥ m1 the following holds. Let x0 ∈ X. Then
(i) If x0 satisfies
(d(x0, pi))
2 ≤ 1
, for some i ∈ {1, . . . , k},
where d( , ) is the distance function on X with respect to ω, then there
is a global holomorphic section sm,x0 of L
m satisfying |sm,x0 |2hm(x0) = 1
(m+ n)!
‖sm,x0‖2L2(hm) =
|Gpi |
+O(m−1).
(ii) If x0 satisfies
< (d(x0, pi))
, for some i ∈ {1, . . . , k},
where am = m/(logm)
2, then there is a global holomorphic section
sm,x0 of L
m satisfying |sm,x0 |2hm(x0) = 1 and
1 + O(m−1)
≤ (m+ n)!
‖sm,x0‖2L2(hm) ≤ C2
1 + O(m−1)
for positive constants C1 and C2 depending only on |Gpi |.
(iii) If x0 satisfies
(d(x0, pi))
, for every i ∈ {1, . . . , k},
then there exists a global holomorphic section sm,x0 of L
m satisfying
|sm,x0 |2hm(x0) = 1 and
(m+ n)!
‖sm,x0‖2L2(hm) = 1 + O(m
Proof We will choose m1 ≫ 1 later in the proof, depending only on X, L
and h. Let m ≥ m1. For (i), assume first that x0 ∈ X − {p1, . . . , pk}. We
may assume without loss of generality that
(d(x0, p1))
2 ≤ 1
and d(x0, pi) ≥ c > 0 for i = 2, . . . , k for some uniform c.
Dropping the subscript p1, we have a uniformizing coordinate system
π : Ṽ → Ṽ /G ∼= V centered at p1 ∈ V , so that at 0 ∈ Ṽ , the metric g is the
identity. The metric is G-invariant and smooth in Ṽ and has vanishing first
derivatives at the origin. Set |G| = l. Since the singularity is isolated, the
only fixed point of the action is 0 ∈ Ṽ , and so Ṽ − {0} is a l-fold cover of
V − {p1}. The preimage of x0 under the map π consists of l distinct points
which we will write as x̃1, . . . , x̃l ∈ Ṽ ⊂ Cn. We may assume that
0 < |x̃1|2 = |x̃2|2 = · · · = |x̃l|2 <
Let η and ψ be the functions defined earlier in the smooth case, and let ψ̃
be a weight function on U given by
ψ̃(z) =
ψ(z − x̃i).
Observe that ψ̃ is G-invariant, since ψ(z) is a function of |z|2 only. Hence ψ̃
can be regarded as a smooth function on X in the orbifold sense. Note also
that ψ̃ is non-positive everywhere. We have
−1∂∂ψ̃(z) ≥ −C(n+ 2)amω(z), for z ∈ Ṽ − {x1, . . . , xl}.
Hence for sufficiently large m, with ψ̃j approximating ψ̃ as before,
−1∂∂ψ̃j −
∂∂ log hm +Ric(ω) ≥
Let v be a local orbifold holomorphic section of L. We may assume without
loss of generality that |v|2h(p1) = 1. Pulling back to Ṽ we have (∂|v|2h)(0) = 0.
Let w be the smooth local section of Lm defined in Ṽ by
am|z − x̃i|2
Notice that w is G-invariant. If m1 is sufficiently large then
|x̃i − x̃j|2 ≤
for all i, j and it follows that w vanishes identically in the regions
|z − x̃i|2 ≤
|z − x̃i|2 ≥
for all i. In addition, w vanishes in the regions
|z|2 ≤ 1
|z|2 ≥ 16
It follows that w descends to a smooth global orbifold section of Lm and ψ̃
is uniformly bounded whenever w is not identically zero. Hence in V ,
|w|2hme−ψ̃ ≤ Cam|vm|2hm .
|w|2hme−ψ̃
and we can apply the orbifold version of Hörmander’s estimates to obtain a
smooth global orbifold section u of Lm satisfying ∂u = w and
|u|2hme−ψ̃
|w|2hme−ψ̃
(logm)2
1/4am≤|z|2≤16/am
|vm|2hm
We can write |v|2h(z) = 1 − |z|2 + O(|z|3) and it follows that, by a similar
argument as in the smooth case,
|u|2hm
≤ C(logm)2n−2m− logm/2−n.
Now set
sm,x0(z) =
am|z − x̃i|2
vm(z)− u(z),
so that sm,x0 is a global holomorphic orbifold section of L
m. Notice that
since
|u|2hme−ψ̃
it follows from the definition of ψ̃ that u(z − x̃i) = O(|z − x̃i|2) for each i.
Hence u(x0) = 0 and since |sm,x0 |2hm(x0) = |vm|2hm(x0), we have
1 ≥ |sm,x0 |2hm(x0) ≥ (1−
)m = em log(1−2/m
2) = 1−O(m−q), (2.8)
for any q. Calculate, remembering that Ṽ −{0} is an l-fold cover of V −{p1},
|sm,x0 |2hm
am|z − x̃i|2
|vm|2hm
am|z − x̃i|2
vm, u
|u|2hm
. (2.9)
The last two terms are O(m−q) for any q. For the first term, observe that
|z|2≤1/4am
|vm|2hm
am|z − x̃i|2
|vm|2hm
|z|2≤16/am
|vm|2hm
. (2.10)
From Lemma 2.2 we obtain
|sm,x0 |2hm
l(m+ n)!
(1 + O(m−1)).
Then from (2.8) we obtain the required section by rescaling.
The case when x0 is one of the singular points pi is easier, since we can
take the weight function to be ψ, which is of course G-invariant. The proof
follows as in the smooth case, except that a factor of l arises when estimating
the integral of |sm,x0 |2hm .
We now consider case (ii). We divide this into two parts:
< (d(x0, p1))
≤ (d(x0, p1))2 <
for a constant A≫ 0 depending on l to be determined later.
For (a), we can use almost the same argument as in (i). The only differ-
ence is that (2.8) becomes
1 ≥ |sm,x0 |2hm(x0) ≥ c > 0,
for a constant c depending on A. The required estimate follows after scaling
sm,x0 .
For (b) we argue as follows. Using the notation above, we work in the
coordinate patch Ṽ and consider the same weight function ψ̃. Let v1 be a
local holomorphic section of π∗L over Ṽ with the property that
|v1|2h(z) = 1− |z − x̃1|2 +O(|z − x̃1|3).
Observe that v1 is not G-invariant. Writing the elements of G as γ1, . . . , γl
with γ1 the identity element, we set vi = γ
i v1, so that
|vi|2h(z) = 1− |z − x̃i|2 +O(|z − x̃i|3).
Now define a local G-invariant section ŵ of Lm over Ṽ by
ŵ(z) =
am|z − x̃i|2
vmi (z).
We have
≤ |x̃i − x̃j |2 ≤
, (2.11)
and by a similar argument as in case (i),
|ŵ|2hme−ψ̃
Hence we can obtain a smooth global orbifold section û of Lm satisfying
∂û = ŵ and
|û|2ω
≤ C(logm)2n−2m− logm/2−n.
Let sm,x0 be the global holomorphic orbifold section of L
m given by
sm,x0(z) =
am|z − x̃i|2
vmi (z)− û(z).
Notice that û(x0) = 0. For i 6= j we have |vmi |2hm(x̃j) ≤ e−A/8, using (2.11),
and if A is sufficiently large depending only on l it follows that
1 ≥ |sm,x0 |2hm(x0) ≥ c > 0, (2.12)
for c depending only on l. Now
|sm,x0 |2hm
am|z − x̃i|2
am|z − x̃i|2
vmi , û
|û|2hm
. (2.13)
As before, the last two terms are O(m−q) for any q, and
(m+ n!)
(1 + O(m−1)) ≤ 1
am|z − x̃i|2
l(m+ n!)
(1 + O(m−1)), (2.14)
for a constant c′ > 0 depending only on l. Combining (2.12), (2.13) and
(2.14) completes part (b) of (ii).
For case (iii) we can avoid the singularities using the same argument as
in the smooth case. �
Remark 2.1 The result of Lemma 2.3.(ii) is clearly not sharp. It would be
interesting to know what the optimal estimates are in this case.
3. Convergence of Tsuji’s iteration
In this section we give a proof of Theorem 1. We begin with a simple and
well-known observation, which we will be useful later. Let X be any set, and
let H be a finite dimensional vector subspace of the vector space of functions
from X to C. Suppose that H is equipped with an inner product 〈 , 〉H . For
any orthonormal basis (v0, . . . , vN ) of H, define a function ρ : X → R by
ρ(x) =
i=0 |vi|2(x). Note that the function ρ is independent of the choice
of orthonormal basis. Now fix x ∈ X . Then it is possible to choose an
orthonormal basis (v0, . . . , vN ) such that vi(x) = 0 for i = 1, 2, . . . , N . The
observation is that
ρ(x) = sup
|v|2(x)
∣ v ∈ H, ‖v‖H = 1
= |v0|2(x). (3.1)
We now turn to the proof of Theorem 1. Notice that, in addition to a
Hermitian metric hm on K
X for each m ≥ m0, we have defined by (1.1) an
inner product 〈 , 〉Tm on H0(X,KmX ) for each m ≥ m0 + 1. Also, from the
Hermitian metric hKE on KX we have the L
2 inner product on H0(X,KmX )
given by hmKE and
ωnKE. We will denote this inner product simply by
〈·, ·〉KE.
We will use Lemma 2.1 on the existence of peak sections to prove the
following:
Lemma 3.1 Let m1 be the constant of Lemma 2.1 for L = KX , h = hKE.
Assume m1 ≥ m0. There exists A depending only on X such that for all
m ≥ m1,
hm1KE
≤ hm ≤ sup
hm1KE
(3.2)
Proof In the course of this proof, the constant A may change from line
to line. We will prove the upper bound on hm first. We use induction.
Obviously, the inequality holds for m = m1. Let
Cm = sup
and assume that hm ≤ CmhmKE. Notice that for any section t of H
0(X,Km+1X ),
‖t‖2Tm+1 ≤ Cm ‖t‖
Fix a point x0 ∈ X. Let sm+1,x0 ∈ H0(X,Km+1X ) be a peak section as
constructed in Lemma 2.1. Then calculate, from the definition of hm+1,
(m+ 1)!
(m+ n+ 1)!
hm+1(x0) ≤
‖sm+1,x0‖
|sm+1,x0 |
hm+1KE (x0)
≤ Cm ‖sm+1,x0‖
hm+1KE (x0)
(m+ 1)!
(m+ n+ 1)!
1 + O
hm+1KE (x0),
and it follows that
hm+1 ≤ sup
We turn now to the lower bound for hm. Again we use induction and assume
that hm ≥ DmhmKE for
Dm = inf
Fix x0 inX. Let (σ
m+1, . . . , σ
(Nm+1)
m+1 ) be an orthonormal basis of H
0(X,Km+1X )
with respect to the inner product 〈 , 〉Tm+1 . We may assume that
m+1(x0) = 0 for i = 1, . . . , Nm+1.
Observe that 1 = ‖σ(0)m+1‖2Tm+1 ≥ Dm‖σ
m+1‖2KE. Then
(m+ 1)!
(m+ n+ 1)!
hm+1(x0) = h
KE (x0)
|σ(0)m+1|2hm+1
≥ Dmhm+1KE (x0)
‖σ(0)m+1‖2KE
|σ(0)m+1|2hm+1
. (3.3)
Now let (τ
m+1, . . . , τ
(Nm+1)
m+1 ) be an orthonormal basis of H
0(X,Km+1X ) with
respect to the inner product 〈 , 〉KE. As before we may assume that
m+1(x0) = 0 for i = 1, . . . , Nm+1.
Then it follows that if t is any section of H0(X,Km+1X ), we have
‖t‖2KE
≤ |τ (0)m+1|
(x0). (3.4)
Hence
(m+ 1)!
(m+ n+ 1)!
hm+1(x0) ≥ Dmhm+1KE (x0)
|τ (0)m+1|2hm+1
. (3.5)
We consider again the peak section sm+1,x0 . Define real numbers a0, . . . , aNm+1
sm+1,x0 =
Then, using the second part of Lemma 2.1,
a2i =
sm+1,x0 ,
‖sm+1,x0‖KE
and so
a2i ≤ A
‖sm+1,x0‖2KE
(m+ 1)2
Now notice that
a20 = ‖sm+1,x0‖2KE −
≥ ‖sm+1,x0‖2KE
1−A 1
(m+ 1)2
Now since |sm+1,x0 |2hm+1
(x0) = 1, we have |τ
m+1|2hm+1
(x0) = 1/a
0. Then
from (3.5) we have
hm+1(x0) ≥ Dmhm+1KE (x0)
and the required lower bound follows. �
From this, we can prove Theorem 1.
Proof of Theorem 1 Raise (3.2) to the power 1/m. For the upper bound
of Theorem 1, observe that
for a constant C depending only on A. The lower bound follows similarly.
4. The modified iteration
We give a proof of Theorem 2. We omit the proof of Theorem 3, since it is
simpler and follows along the same lines. We first prove a convergence result
on the minimal surface Xmin of general type. Consider a Hermitian metric
hm0 on K
and write β = |S−2|2hC , for |S−2|
as in the introduction.
Recall that hmin is the Hermitian metric on KXmin given by f
∗hKE, for hKE
the Hermitian metric on KXcan corresponding to the Kähler-Einstein metric
ωKE. Consider the sequence of metrics hm,ε = hm,ε(β, hm0) on K
Theorem 4.1 For every sequence εj → 0,
lim sup
h1/mm,εj → hmin, as j → ∞,
almost everywhere on Xmin.
To prove this, we will need two lemmas.
Lemma 4.1 There exist m1 > 0 and A depending only on Xmin, β and ε
such that for all m ≥ m1,
βεhm,ε ≤ sup
hm1,ε
hmmin
. (4.1)
Proof We will use induction, for m1 to be determined later. Assume the
inequality holds for hm. Let
Cm = sup
hm1,ε
hm1min
Then βεhm,ε ≤ Cmhmmin. It follows that ‖t‖2Tm+1,ε ≤ Cm‖t‖
for t any
global section of Km+1Xmin . Since the inequality for hm+1,ε obviously holds at
points on C, it is sufficient to prove it at a fixed point y0 ∈ Xmin − C. Write
x0 = f(y0) ∈ Xcan. By Lemma 2.3 there is a global holomorphic section
sm+1,x0 of K
satisfying |sm+1,x0 |2hm+1
(x0) = 1 and
βε(y0)
|sm+1,x0 |2hm+1
(m+ 1)!
(m+ n+ 1)!
1 + O(m−1)
as long as m1 is chosen to be sufficiently large. Then
(m+ 1)!
(m+ n+ 1)!
βε(y0)hm+1,ε(y0)
≤ βε(y0)
‖f∗(sm+1,x0)‖
Tm+1,ε
|f∗(sm+1,x0)|
hm+1min (y0)
≤ Cmβε(y0)
|sm+1,x0 |2hm+1
hm+1min (y0)
(m+ 1)!
(m+ n+ 1)!
1 + O
hm+1min (y0),
and the lemma follows. �
For the lower bound of hm,ε, we use a modification of a lemma of Tsuji
[Ts]:
Lemma 4.2 There exist constants m2 and B depending only on Xmin such
that for all m ≥ m2 and 0 < ε ≤ 1,
βεh−1/mm,ε ≤ V
m−m2+1
βεh−1/m2m2,ε
)m2/m
(4.2)
for V =
ωnmin
Proof Set
Lm,ε =
βεh−1/mm,ε
(m+ n)!
(Nm + 1),
for Nm + 1 = dimH
0(Xmin,K
). Then we claim that
Lm,ε ≤ c1/mm L
(m−1)/m
m−1,ε . (4.3)
Given (4.3), we can finish the proof of the lemma as follows. First, by
Riemann-Roch, there exist constants m2 and B such that for m ≥ m2,
cm ≤ V
From (4.3), arguing by induction, we have
Lm,ε ≤ (cmcm−1 · · · cm2)
Lm2/mm2,ε ,
and the inequality (4.2) follows immediately. It remains to show (4.3). Using
Hölder’s inequality,
Lm,ε =
βεh−1/mm,ε h
1/(m−1)
m−1,ε h
−1/(m−1)
m−1,ε
βε(h−1/mm,ε h
1/(m−1)
m−1,ε )
−1/(m−1)
m−1,ε
−1/(m−1)
m−1,ε
βεh−1m,εhm−1,ε
(m−1)/m
m−1,ε
(m+ n)!
σ(i)m,ε ⊗ σ
m,ε ⊗ hm−1,ε
(m−1)/m
m−1,ε
= c1/mm L
(m−1)/m
m−1,ε ,
and this completes the proof of the lemma. �
We can now use these lemmas to prove a convergence result for the
metrics hm,ε.
Proof of Theorem 4.1 From Lemma 4.1 we have
h−1/mm,ε ≥ E(m, ε)h−1min on Xmin − C,
where E(m, ε) → 1 as m→ ∞. From Lemma 4.2, we have
βεh−1/mm,ε ≤ V F (m, ε),
where F (m, ε) → 1 as m→ ∞. Writing hε = lim supm→∞ h
m,ε , we have
h−1ε − h−1min
h−1min
h−1min =
lim inf
βε(h−1/mm,ε − E(m, ε)h−1min)
≤ lim inf
βεh−1/mm,ε −
βεh−1min
≤ V −
βεh−1min → 0,
as ε→ 0. Theorem 4.1 follows. �
Finally, we complete the proof of Theorem 2.
Proof of Theorem 2 Using the notation given in the introduction, there
is an isomorphism Θ : H0(Xmin,K
) → H0(X,KmX ) given by Θ(s) =
τ∗s⊗ Sm−1. Then, given an inner product Tm,ε on H0(X,KmX ), we can define
an inner product T̂m,ε on H
0(Xmin,K
) by 〈s, t〉
T̂m,ε
= 〈Θ(s),Θ(t)〉Tm,ε .
Then given an initial Hermitian metric hm0 on K
X we can obtain an
inner product 〈·, ·〉
T̂m0+1,ε
on Km0+1Xmin and hence a Hermitian metric ĥm0+1,ε
. Applying the modified Tsuji iteration as in the case of Theorem
4.1, we obtain a sequence of Hermitian metrics ĥm,ε form ≥ m0+1 onKmXmin.
From the definition of S−1, one can check that hm,ε = τ
∗ĥm,ε ⊗ |S−1|−2m.
Indeed, assuming inductively that hm,ε = τ
∗ĥm,ε ⊗ |S−1|−2m, denote by
〈·, ·, 〉′
T̂m+1,ε
the inner product induced by ĥm,ε on H
0(Xmin,K
). We need
to show that 〈s, t〉′
T̂m+1,ε
= 〈s, t〉
T̂m+1,ε
for s, t ∈ H0(Xmin,Km+1Xmin). But
〈s, t〉′
T̂m+1,ε
ĥm,ε ⊗ s⊗ t
hm,ε ⊗ |S−1|2m ⊗ (τ∗s)⊗ (τ∗t)⊗ |S−1|2
= 〈Θ(s),Θ(t)〉Tm+1,ε
= 〈s, t〉
T̂m+1,ε
Now by Theorem 4.1, we see that for any sequence εj → 0, we have
lim sup
ĥ1/mm,εj → hmin,
almost everywhere. Theorem 2 follows immediately. �
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