Partial C 0-estimate for Kähler-Einstein metrics
Gang Tian∗
Department of Mathematics
Beijing University and Princeton University
Contents
1 Introduction
1
2 Proving Theorem 1.5
3
3 Proving Theorem 1.7
8
4 Extension to conic Kähler-Einstein metrics
1
10
Introduction
In this paper, we give a proof of my conjecture on the partial C 0 -estimate for
Kähler-Einstein metrics with positive scalar curvature. As a corollary, as I
already pointed out in [Ti09], the Gromov-Hausdorff limits of Kähler-Einstein
metrics are projective varieties.
Let M be a compact manifold which admits Kähler metrics with its first
Chern class c1 (M ) positive. We denote by K(M ) the set of all Kähler metrics
ω on M with Kähler class [ω] = c1 (M ). Consider its set
K(M, t0 ) = { ω ∈ K(M ) | Ric(ω) ≥ t0 ω }.
Clearly, K(M, t0 ) is empty unless t0 ≤ 1 and K(M, 1) is the set of KählerEinstein metrics on M with Kähler class c1 (M ).
By the Kodaira embedding theorem, for ℓ sufficiently large, any basis of
−ℓ
H 0 (M, KM
) embeds M into a projective space CP N . For any ω ∈ K(M, t0 ),
choose a Hermitian metric h with ω as its curvature form and any orthonormal
−ℓ
basis {Si }0≤i≤N of each H 0 (M, KM
) with respect to the induced inner product
by h and ω. Put
N
∑
ρω,ℓ (x) =
||Si ||2h (x).
(1.1)
i=0
This is independent of the choice of h and the orthonormal basis {Si }.
∗ Supported
partially by a NSF grant
1
Conjecture 1.1. [Ti90] There are uniform constants ck = c(k, n) > 0 for k ≥ 1
and ℓi → ∞ such that for any ω ∈ K(M, t0 ) and ℓ = ℓi for each i,
ρω,ℓ ≥ cℓ > 0.
(1.2)
Remark 1.2. In fact, I expect the stronger version of Conjecture 1.1: There
are uniform constants ck = c(k, n) > 0 for k ≥ 1 and ℓ0 = ℓ0 (n) such that for
any ω ∈ K(M, t0 ), and ℓ ≥ ℓ0 , ρω,ℓ ≥ cℓ .
The resolution of this conjecture will lead to a proof of the Yau-TianDonaldson conjecture: If M is K-stable for all sufficiently large ℓ, then M
admits a Kähler-Einstein metrics.
If ωi is a sequence of Kähler metrics on M with [ωi ] = c1 (M ) and their
Ricci curvature greater than or equal to t0 > 0, then by taking a subsequence if
necessary, we may assume that (M, ωi ) converge to a length space (M∞ , d∞ ).
On the other hand, for ℓ sufficiently large, we have embeddings σi : M ,→
−ℓ
CP N by an orthonormal basis of H 0 (M, KM
) with respect to ωi . By taking
a subsequence if necessary, we may assume that σi (M ) ⊂ CP N converge to a
holomorphic cycle M̄∞ ⊂ CP N . It was known (see [Ti09]) that the irreducibility
of M̄∞ implies Conjecture 1.1.
I have expected since early 90’s:
Conjecture 1.3. The Gromov-Hausdorff limit M∞ coincides with the complex
limit M̄∞ . In particular, M̄∞ is irreducible
Our main theorem of this paper is to confirm Conjecture 1.1 for K(M, 1),
precisely,
Theorem 1.4. There are a positive constant ϵ = ϵ(n) > 0 and sufficiently
large ℓ = ℓ(n) such that ρω,ℓ ≥ ϵ for all ω ∈ K(M, 1).
It is known that for each n, there are only finitely many family of compact
Kähler manifolds of complex dimension n and with positive first Chern class.
Hence, Theorem 1.4 holds for all Kähler-Einstein manifolds of dimension n.
Let (Mi , ωi ) be any sequence of Kähler-Einstein manifolds with Ric(ωi ) = ωi
and which converges to (M∞ , d∞ ) in the Gromov-Hausdorff topology. Theorem
1.4 follows from the following:
Theorem 1.5. There are a positive constant ϵ = ϵ(n) > 0 and sufficiently
large ℓ = ℓ(n) such that ρωi ,ℓ ≥ ϵ for all (Mi , ωi ).
It follows from [CCT95] that there is a closed subset S ⊂ M∞ of Hausdorff
codimension at least 4 such that M∞ \S is a smooth Kähler manifold and d∞
is induced by a Kähler-Einstein metric ω∞ outside S with Ric(ω∞ ) = ω∞ .
Moreover, ωi converges to ω∞ in the C ∞ -topology outside S.
A consequence of the above theorem implies (as indicated in [Ti09])
Theorem 1.6. The Gromov-Hausdorff limit M∞ is a variety embedded in some
CP N and S is a subvariety.
2
This theorem affirms Conjecture 1.3 for Kähler-Einstein metrics. The proof
of this theorem is based on the same arguments as those in the proof of Theorem
1.4, that is, constructing holomorphic sections which separate points for the limit
of embeddings σi . We will omit the details in this note.
We can extend Theorem 1.5 to the case of almost Kähler-Einstein metrics.
Let ωi be a sequence of Kähler metrics on M with [ωi ] = c1 (M ) and Ric(ωi ) ≥
(1−ϵi )ωi , where ϵi ≥ 0 and lim ϵi = 0. Assume that (M, ωi ) converge to a metric
space (M∞ , d∞ ) in the Gromov-Hausdorff topology. It is proved in [TW11] that
M∞ is smooth outside a closed subset S of Hausdorff codimension at least 4
and the restriction of d∞ to M∞ \S is given by a Kähler-Einstein metric ω∞ .
We can prove the following
Theorem 1.7. Let (M∞ , ω∞ ) be as above. Then there is a sufficiently large ℓ
such that ρω∞ ,ℓ > 0.
The proof of this is based on the regularity theory in [TW11] and an extension of the L2 -estimate to singular spaces like (M∞ , ∞).
Theorem 1.4 and 1.6 were announced with an outlined proof in our expository paper for the proceeding of Calabi’s 85th birthday edited by Bourguignon,
Chen and Donaldson. In the next section of this note, we provide a proof of
the first theorem following the arguments in [Ti12]. The second theorem follows
easily as we indicated above. During the preparation of this note, we learned
that Donaldson and Sun [DS12] also gave an independent and different proof
of Theorem 1.4, though the two proofs share some overlapping ideas which appeared in previous works. One can also find a proof of Theorem 1.6 in [DS12].
In Section 3, we give a proof for Theorem 1.7. In Section 4, we discuss the
extension of Theorem 1.4 to conic Kähler-Einstein metrics. An outlined proof
will be given while details will appear later.
2
Proving Theorem 1.5
The proof of Theorem 1.5 is essentially a localized version of the proof for the
following:
Proposition 2.1. By taking a subsequence if necessary, for each ℓ, we have
−ℓ
−ℓ
that H 0 (Mi , KM
) converges to H 0 (M∞ , KM
) as i tends to ∞ in the sense:
i
∞
−ℓ
There are orthonormal bases {σai }0≤a≤N of H 0 (Mi , KM
) with respect to hi
i
i
∞
such that σa converges to σa (0 ≤ a ≤ N ) as i tends to ∞ and {σa∞ } forms an
−ℓ
orthonormal basis of H 0 (M∞ , KM
).
∞
¯
As in [Ti89], we prove this by using the L2 -estimate for ∂-operator
and the
theory for elliptic equations. Next we recall from [Ti90]:
−ℓ
Lemma 2.2. For each i and any σ ∈ H 0 (Mi , KM
), we have the following
i
identities:
∆ωi ||σ||2 = ||∇σ||2 − nℓ ||σ||2
(2.1)
3
and
∆ωi ||∇σ||2 = ||∇2 σ||2 − ((n + 2) ℓ − 1) ||∇σ||2 .
(2.2)
−ℓ
)
Corollary 2.3. There is a uniform constant C0 such that for any σ ∈ H 0 (Mi , KM
i
(ℓ > 0), we have
(∫
sup ||σ||
≤
C0 ℓ
n
2
Mi
sup ||∇σ||
||σ||
2
ωin
) 12
Mi
≤
C0 ℓ
n+1
2
Mi
(∫
||σ||
2
ωin
(2.3)
) 12
.
(2.4)
Mi
It follows from Lemma 2.2 and the standard Moser iteration since the Sobolev
constants of (Mi , ωi ) are uniformly bounded due to some results of C. Croke
and P. Li (see [Ti87] and its references).
It follows that by taking a subsequence if necessary, we may assume σai
converges to a σa∞ as i tends to ∞. Furthermore, one can show that ρωi ,ℓ are
uniformly continuous and converge to ρω∞ ,ℓ which is also continuous on M∞ .
Thus, in order to prove Theorem 1.5, we only need to show
inf ρω∞ ,ℓ (x) > 0.
x
(2.5)
Since ρω∞ ,ℓ is continuous and M∞ is compact, it suffices to show that for
any x ∈ M∞ , there is ℓ = ℓx such that
ρω∞ ,ℓ (x) > 0.
(2.6)
This can be achieved by using the L2 -estimate and the structure results on
M∞ from [CCT95].
According to [CCT95], for any ri 7→ 0, by taking a subsequence if necessary, we have a tangent cone Cx of (M∞ , ω∞ ) at x, where Cx is the limit
limi→∞ (M∞ , ri−2 ω∞ , x) in the Gromov-Hausdorff topology, satisfying:
1. Cx is a Kähler cone with vertex o;
2. Each Cx is regular outside a closed subcone Sx of complex codimension at
least 2. Such a Sx is the singular set of Cx ;
3. There is an natural Kähler Ricci-flat metric gx on Cx \Sx which is also a cone
metric.
√
¯ 2 on
Since gx is a Kähler cone metric, its Kähler form ωx is equal to −1∂ ∂ρ
x
the regular part of Cx , where ρx denotes the distance function from the vertex of
Cx , denoted by x for simplicity. In other words, the trivial bundle Lx = Cx × C
2
over Cx admits a Hermitian metric e−ρx | · |2 whose curvature is ωx .
Without loss of generality, we may choose ri such that ki = ri−2 are integers.
Now we fix some notations: For any r > 0 and 0 < δ < ϵ, we denote by
V (x; δ, ϵ, r) the set
{sv ∈ Cx | s ∈ (δ, r), v ∈ ∂B1 (x, gx ), d(v, Sx ∩ ∂B1 (o, gx )) > ϵ},
where BR (o, gx ) denotes the geodesic ball of (Cx , gx ) centered at the vertex and
with radius R.
4
If Cx has isolated singularity, i.e., ∂B1 (x) is smooth, then we can drop ϵ and
write
V (x; δ, r) = {(sv) ∈ Cx | s ∈ (δ, r), v ∈ ∂B1 (x)}.
Let ki be the above sequence such that (M∞ , ki ω∞ , x) converges to (Cx , gx , x).
By [CCT95], for any given δ, ϵ > 0, whenever i is sufficiently large, there are
diffeomorphism ϕi : V (x; δ, ϵ, 2) 7→ M∞ \S, where S is the singular set of M∞ ,
satisfying:
(1) d(x, ϕi (V (x; δ, ϵ, 2))) < δri and ϕi (V (x; δ, ϵ, 2)) ⊂ B3ri (x), where BR (x) the
geodesic ball of (M∞ , ω∞ ) with radius R and center at x;
(2) If g∞ is the Kähler metric with the Kähler form ω∞ on M∞ \S, then
lim ||ri−2 ϕ∗i g∞ − gx ||C 6 (V (x;δ/2,ϵ/2,3)) = 0,
i→∞
(2.7)
where the norm is defined in terms of the metric gx .
Lemma 2.4. There is an integer a > 0 such that for i sufficiently large, there
−aki
are δi 7→ 0 1 and isomorphisms ψi from the trivial bundle Cx × C onto KM
∞
over V (x; δ, ϵ, 2) commuting with ϕi satisfying:
||ψi (1)||2 = e−ρx
2
and
||Dψ||C 4 ≤ δi ,
(2.8)
−ki
where || · || denotes the induced norm on KM
by gx , D denotes the covariant
∞
−ρ2x /2
derivative with respect to the norms || · || and e
| · |.
Proof. First we note that by a recent result of Colding-Naber [CN10], the regular
g x ) is a
part Reg(Cx ) of Cx is geodesically convex, so its universal cover Reg(C
finite cover, say of the order a ≥ 1. Since gx is a Kähler Ricci-flat metric on Cx ,
g x has its holonomy group in SU(n) and consequently,
the induced metric on RegC
−a
−a
KCx admits a parallel section. It follows that for i sufficiently large, KM
is
∞
trivial over ϕi (V ) for any open subset V whose closure is contained in Reg(Cx ).
This makes it plausible to construct the isomorphisms ψi . For simplicity, we
assume a = 1 or equivalently, Reg(Cx ) is simply-connected, otherwise, we do the
g x ). We cover V (x; δ, ϵ, 2) by finitely many
following on its universal cover Reg(C
geodesic balls Bsα (yα ) (1 ≤ α ≤ N ) such that the closure of each B2sα (yα ) is
strongly convex and contained in Reg(Cx ). Now we construct ψi . For simplicity
of notations, we fix a sufficiently large i and write r = ri , k = ki etc.. First we
construct ψ̃α over each B2sα (yα ). For any y ∈ B2sα (yα ), let γy ⊂ B2sα (yα ) be
the unique minimizing geodesic from yα to y. We define ψ̃α as follows: First we
2
−k
define ψ̃α (1) ∈ Li |ϕ(yα ) , where Li = KM
, such that ||ψ(1)||2 = e−ρx (yα ) . Next,
∞
for any y ∈ Uα , where Uα = B2sα (yα ), define ψ̃α : C 7→ Li |y by ψ̃α (a(y)) =
τ (ϕ(y)), where a(y) is the parallel transport of 1 along γy with respect to the
standard norm above and τ (ϕ(y)) is the parallel transport of ψ(1) along ϕ · γy .
Clearly, we have the first equation in (2.8). The estimates on derivatives can be
1 In
fact, δi can be chosen depending only on ||ki ϕ∗ g∞ − gx ||C 6 (V (x;δ/2,ϵ/2,3)) .
5
done as follows: If a : Uα 7→ Uα × C and τ : Uα 7→ ϕ∗ L|Uα are two sections such
that ψ̃α (a) = τ , then we have the identity: Dτ = Dψ̃α (a) + ψ̃α (Da), where D
denote the covariant derivatives with respect to various norms on line bundles.
By the definition, one can easily see that Dψ̃α (yα ) ≡ 0. To estimate Dψ̃α at y,
we differentiate along γy to get
DT DX τ = DT (DX ψ̃α (a)) + ψ̃α (DT DX a),
where T is the unit tangent of γy and X is a vector field along γy with [T, X] = 0.
Here we have used the fact that DT ψ̃α = 0 which follows from the definition.
Using the curvature formula, we see that it is the same as
ki ϕ∗ ω∞ (T, X)ψ̃α (a) = DT (DX ψ̃α (a)) + ωx (T, X)a.
Using
lim ki ϕ∗ ω∞ = λi ϕ∗ ω∞ = ωx ,
i→∞
we deduce from the above that DT (DX ψ̃α (a)) converges to 0 as i tends to ∞.
Since DX ψ̃α = 0 at yα , we see that ||Dψ̃α ||C 0 (U ) can be made sufficiently small.
The higher derivatives can be bounded in a similar way.
Next we want to modify each ψ̃α to get the required ψ = ψi . For any α, β,
we set
θαβ = ψ̃α−1 ◦ ψ̃β : Uα ∩ Uβ 7→ S 1 .
Clearly, θαγ = θαβ · θβγ on Uα ∩ Uβ ∩ Uγ , so we have a closed cycle {θαβ }. Using
−k
the fact that KM
is trivial over ϕi (V ) for any V b Reg(Cx ), one can deduce
∞
from the simply-connectedness that this cycle has to be exact, moreover, by
replacing each Uα by Bsα (xα ), we can construct ζα : Bsα (xα ) 7→ S 1 satisfying:
1. ψ̃α · ζα = ψ̃β · ζβ on Bsα (xα ) ∩ Bsβ (xβ );
2. ||Dζα ||C 3 (Bsα (xα )) is dominated by ||Dψ̃α ||C 3 (U ) .
Then we can get ψ by setting ψ = ψ̃α · ζα on V (x; δ, ϵ, 2) ∩ Bsα (xα ). It is easy
to show that this ψ has all the properties we asked.
We also have the following extension property:
Lemma 2.5. Given any c > 0, there is a constant C, which may depend on
c and the cone Cx 2 , such that for any holomorphic function f on V (x; δ, ϵ, 2)
with |f |, |df | ≤ c and f ≥ 1 on ∂B1 (o, gx ) ∩ V (x; δ, ϵ, 2), we have
√
f ( δv) ≥ 1 − Cϵ2 log δ,
1
where v ∈ ∂B1 (o, gx ) and d(v, Sx ∩ ∂B1 (x, gx ) ≥ 8ϵ n .
2 It is possible to remove the dependence on C ′ by using known estimates on Green function
x
over manifolds with non=negative Ricci curvature.
6
Proof. This is proved by using the Green function on B1 (x, gx ) with boundary
value 0. Let η be a cut-off function satisfying: η(t) = 0 for t ≤ 1, η(t) = 1 for
t ≥ 2 and |η ′ (t)| ≤ 1. Define
F (sv) = η(δ −1 s) η(ϵ−1 d(v, Sx ∩ ∂B1 (x, gx ))) f (sv),
where s ∈ (0, 1) and v ∈ ∂B1 (x, gx ). Then F vanishes near the singularity of Cx
and smooth on Reg(Cx ). Clearly, ∆F supports in V (x; δ, ϵ, 2)\V (x; 2δ, 2ϵ, 2),
moreover, for some uniform constant C ′ , we have
|∆F | ≤ C ′ s−2 ϵ−2 on V (x; δ, ϵ, 2)\ (V (x; 2δ, 2ϵ, 2) ∪ B4δ (0, gx ))
and
|∆F | ≤ C ′ δ −2 on B4δ (o, gx )\Bδ (o, gx ).
If G(·, ·) denotes the Green function on B1 (o, gx ) with boundary value 0, then
we have
∫
∫
∂G(y, z)
f (y) =
F (z)
dz −
∆F (z)G(y, z)dz.
(2.9)
∂ν
∂B1 (o,gx )
B1 (o,gx )
Note that for any y =
√
δv as given and z ∈ V (x; δ, ϵ, 2)\V (x; 2δ, 2ϵ, 2), we have
|G(y, z)| ≤ C ′′ min{δ −(n−1) , s−2(n−1) }
for some constant C ′′ . Since Sx is codimension at least 4 and the dimension of
Cx is at least 4, we can deduce from the above estimates
∫
∆F (z)G(y, z)dz ≤ Cϵ2 log δ.
B1 (o,gx )
Then the lemma follows easily from (2.9).
Now we can apply the L2 -estimate to proving (2.6), and consequently, Theorem 1.5. Fix 0 < δ < ϵ small and to be determined later. For each i suffi−ki
ciently large, there is a section τi = ψi (e) of KM
on ϕi (V (x; δ, ϵ, 2)) such that
∞
2
2
1−ρx
||τi || = e
. Clearly, ||τi || is greater than 1 inside B1 (x) and less than 1
¯ i || ≤ Cδi for some uniform constant C. Let
outside B1 (x). By Lemma 2.4, ||∂τ
η be the cut-off function satisfying: η(t) = 0 for t ≤ 1, η(t) = 1 for t ≥ 2 and
|η ′ (t)| ≤ 1. We define for any y = sv ∈ V (x; δ, ϵ, 2)
τ̃i (ϕi (y)) = η(7/2 − s) η(δ −1 s) η(ϵ−1 d(v, Sx ∩ ∂B1 (o, gx ))) τ l (ϕi (y)).
(2.10)
Here l is a large integer which depends only on ϵ. Then by choosing ϵ sufficiently
−lki
small, one can easily show τ̃i extends to a Lipschitz section of KM
on M∞
∞
satisfying:
(i) τ̃i coincides with τiℓ on ϕi (V (x; 2δ, 2ϵ, 3/2));
∫
n
(ii) M∞ ||∂¯τ̃i ||2 ω∞
≤ Cri2n−2 , where C denotes a uniform constant.
7
Note that C, C ′ et al always denote uniform constants. Set ℓ = lki . By the
−ℓ
3
¯ i = ∂¯τ̃i
¯
such that ∂v
L -estimate for ∂-operator
, we get a section vi of KM
∞
and
∫
∫
1
n
n
||vi ||2 ω∞
||∂¯τ̃i ||2 ω∞
≤
≤ Cri2n−2 ℓ−1 .
ℓ M∞
M∞
2
−ℓ
Then σi = τ̃i − vi is a holomorphic section of KM
. By (i),
∞
¯ i = 0
∂v
on ϕi (V (x; 2δ, 2ϵ, 3/2)),
then by applying the standard elliptic estimates, we can get
∫
n
sup
||vi ||2 ≤ C(ϵri )−2n
||vi ||2 ω∞
≤ C ′ ϵ−2n ri−2 ℓ−1 .
ϕi (V (x;2δ,2ϵ,3/2)∩∂B1 (o,gx )
M∞
Choosing l = 4C ′ ϵ−2n , we can show ||σi || ≥ 1/2 on ϕi (V (x; 2δ, 2ϵ, 1)∩∂B1 (o, gx )).
On the other hand, it is clear that ||σi || are uniformly bounded, hence, as i tends
to ∞, σi restricted to ϕi (B3/2 (o, gx ))\S converges to a holomorphic function f
on B3/2 (o, gx ) with f ≥ 1/2 on ∂B1 (o, gx ). Then it follows from Lemma 2.5
√
that ||σi ||(ϕi ( δv) ≥ 1/2 − Cϵ2 log δ for some v ∈ ∂B1 (o, gx ).
By applying the second estimate in Corollary 2.3 to σi , we get
∫
n+1
n
≤ C̃ϵ−n(n+1) ri−1 .
||σi ||2 ω∞
sup ||∇σi || ≤ C ′′ ℓ 2
M∞
M∞
√
√
Since the distance d(xi , ϕi ( δv)) is less than 2 δri for i sufficiently large, we
deduce from the above estimates
√
||σi ||(xi ) ≥ 1/4 − Cϵ2 log δ − 2C̃ϵ−n(n+1) δ,
√
hence, if we choose 0, δ < ϵ satisfying: 32C̃ δ = ϵn(n+1) and 16Cϵ2 log δ < 1,
then ρω∞ ,ℓ (x) > 1/8. The theorem is proved.
3
Proving Theorem 1.7
Clearly, Theorem 1.7 can be proved by the arguments in proving Theorem 1.5
in last section once we prove the following lemma.
Lemma 3.1. Let (M∞ , ω∞ ) be a compact metric space such that ω∞ is a smooth
Kähler-Einstein metric on M∞ \S for some closed subset S of Hausdorff codi−1
mension at least 4 and is the curvature of a Hermitian metric on KM
. Then
∞
−ℓ
for any smooth section v of KM∞ with support outside S, there is a unique
¯ = ∂v
¯ and
section w such that ∂w
∫
∫
1
n
¯ 2 ωn .
||∂v||
||w||2 ω∞
≤
∞
ℓ
M∞
M∞
3 We
can do it directly on M∞ by using the L2 -estimate on Mi and Proposition 2.1.
8
Remark 3.2. For (M∞ , ω∞ ) from last section, this lemma is trivially true.
This is because (M, ωi ) converge to (M∞ \S, ω∞ ) in the smooth topology and
¯
we have the L2 -estimate for the ∂-operator.
However, we do not have such an
approximation by smooth manifolds with Ricci curvature bounded from below.
Proof. We outline a proof here. First we observe the Bochner identity on
−ℓ
Ω0,1 (KM
:
∞
¯ = ∇∇
¯ + ∇∇
¯ + (2ℓ + 3).
2(∂¯∂¯∗ + ∂¯∗ ∂)
Thus we can minimize the functional
∫
(
)
¯ u) ω n
¯ 2 + (2ℓ + 3)||u||2 + 4(∂v,
E(u) =
||∇u||2 + ||∇u||
∞
M∞
among all the sections with finite H 1,2 -norm. It is easy to show that the minimizer, say u, exists and satisfies the equation
¯ = ∂v
¯
∂¯∂¯∗ u + ∂¯∗ ∂u
on M∞ \S,
moreover, such a minimizer u is smooth outside S and
∫
∫
(
)
¯ 2 + ||∂¯∗ u||2 + ||u||2 ω n ≤
||∂u||
∞
M∞
M∞
¯ 2 ωn .
||∂v||
∞
Put w = ∂¯∗ u, then we have
¯
¯
(∂¯∂¯∗ + ∂¯∗ ∂)w
= ∂¯∗ ∂v.
It follows
¯
−∆||w|| ≤ C||w|| + ||∂¯∗ ∂v||.
On the other hand, we have
∫
M∞
n
< ∞.
||w||2 ω∞
Hence, by the standard Moser iteration, we can show that ||w|| is bounded on
M∞ . Then we can do integration by parts and get from the equation on w
∫
∫
∫
1
n
¯ 2 ωn ≤ 1
¯ 2 ωn .
||w||2 ω∞
≤
||∂w||
||∂v||
∞
∞
ℓ
ℓ
M∞
M∞
M∞
The lemma is proved.
If (M∞ , ω∞ ) is the Gromov-Hausdorff limit of (M, ωi ) with [ωi ] = c1 (M ) and
Ric(ωi ) ≥ ti ωi for some ti → 1, then we hope to deduce from Theorem 1.7 that
ρωi ,ℓ ≥ ϵ for a sufficiently large ℓ and a sufficiently small ϵ > 0. We still have the
analogous version of Lemma 2.2 and consequently, the estimates in Corollary
2.3. It follows that ρωi ,ℓ are uniformly continuous and converge to a continuous
limit function on M∞ , however, this limit may not coincide with ρω∞ ,ℓ since we
do not know an analogue of Proposition 2.1 for almost Kähler-Einstein metrics,
or more generally, for Kähler metrics with Ricci curvature bounded from below.
9
Conjecture 3.3. For each t0 > 0 and sufficienly large ℓ, ρω,ℓ (x) is a uniform
continuous function on (ω, x) ∈ K(M, t0 ) × M .
This is a metric version of the flatness for varieties in algebraic geometry.
The above conjecture has an important application to establishing the existence
of Kähler-Einstein metrics. If Conjecture 3.3 is affirmed, we can deduce from
Theorem 1.7
Theorem 3.4. Let M be a compact Kähler manifold with c1 (M ) > 0. Assume
that the K-energy associated to the Kähler class c1 (M ) is bounded from below
−1
and (M, KM
) is K-stable. Then M admits a Kähler-Einstein metric.
In fact, we only need a weak version of Conjecture 3.3: In order to prove
Theorem 3.4, we only need to prove lim ρωi ,ℓ = ρω∞ ,ℓ for a sequence
of al√
¯ i with
most Kähler-Einstein metrics ωi satisfying: Ric(ωi ) − ωi = −1∂ ∂h
lim ||hi ||C 0 = 0.
4
Extension to conic Kähler-Einstein metrics
The theory of smooth Kähler-Einstein metrics can be generalized to the metrics
with conic angle along a divisor. For simplicity, here we consider only the case
of smooth divisors4 .
Let M be a compact Kähler manifold and D ⊂ M be a smooth divisor. A
conic Kähler metric on M with angle 2πβ (0 < β ≤ 1) along D is a Kähler
metric on M \D that is asymptotically equivalent along D to the model conic
metric


n
∑
√
d
∧
dz̄
1
1
ω0,β = −1 
+
dzj ∧ dz̄j  ,
|z1 |2−2β
j=2
where z1 , z2 , · · · , zn are holomorphic coordinates such that D = {z1 = 0} locally.
Each conic Kähler metric can be given by its Kähler form ω which represents a
cohomology class in H 1,1 (M, C) ∩ H 2 (M, R), referred as the Kähler class [ω]. A
conic Kähler-Einstein metric is a conic Kähler metric which are also Einstein.
In this section, we discuss the generalizations of Theorem 1.4 and Theorem
1.6 to conic Kähler-Einstein metrics of positive scalar curvature. Let us first
describe our main results in this section. Let M be a Fano manifold and D be
a smooth divisor which represents the Poincare dual of λc1 (M ). We call ω a
conic Kähler-Einstein if its Kähler class equals to 2πc1 (M ) and satisfies:
Ric(ω) = µω + (1 − β)[D].
(4.1)
Here the equation on M is in the sense of currents, while it is classical outside
D. We require µ > 0 which is equivalent to (1−β)λ < 1. As in the smooth case,
each conic Kähler metric ω with [ω] = 2πc1 (M ) is the curvature of a Hermitian
−1
metric || · || on the anti-canonical bundle KM
. The difference is that it is
4 The
results in this section still hold for divisors with normal crossings.
10
not smooth, but it is Hölder continuous. S. Donaldson suggested a continuity
method of constructing a Kähler-Einstein metric on M by using conic KählerEinstein metrics. It boils down to solving the following complex Monge-Ampere
equations:
√
¯ n = ehβ −µφ ω n ,
(ωβ + −1∂ ∂φ)
(4.2)
β
where ωβ is a suitable family of conic Kähler metrics with [ωβ ] = 2πc1 (M ) and
cone angle 2πβ along D and hβ is determined by
∫
√
¯
(ehβ − 1)ωβn = = 0.
Ric(ωβ ) = µω + (1 − β)[D] + −1∂ ∂hβ and
M
As shown in [JMR11], if µ > 0 is sufficiently small, (4.2) is solvable, so there
is a conic Kähler-Einstein metric with corresponding cone 2πβ along D. Furthermore, as shown in [JMR11], it is crucial in solving (4.2) to establish the a
prior C 0 -estimate for its solutions. Such a C 0 -estimate does not hold in general.
Therefore, as shown in my program on the existence of Kähler-Einstein metrics through the standard continuity method [Ti09], we first establish a partial
C 0 -estimate and then use the K-stability to conclude the C 0 -estimate, consequently, the existence of Kähler-Einstein metrics on Fano manifolds which are
K-stable.
For any t0 > 0 and β0 > 0, let K(M, D, t0 , β0 ) be the set of conic Kähler
metrics with Ricci curvature bounded from below by t0 > 0 and cone angle 2πβ
along D for some 1 ≥ β ≥ β0 . For any ω ∈ K(M, t0 ), choose a C 1 -Hermitian
metric h with ω as its curvature form and any orthonormal basis {Si }0≤i≤N of
−ℓ
each H 0 (M, KM
) with respect to the induced inner product by h and ω. As
before, we have a well-defined function
ρω,ℓ (x) =
N
∑
||Si ||2h (x).
(4.3)
i=0
Conjecture 4.1. There are uniform constants ck = c(k, n, t0 , β0 ) > 0 for k ≥ 1
and ℓi → ∞ such that for any ω ∈ K(M, D, t0 , β0 ) and ℓ = ℓi for each i,
ρω,ℓ ≥ cℓ > 0.
(4.4)
The following confirms Conjecture 4.1 for conic Kähler-Einstein metrics with
angle bounded from below.
Theorem 4.2. There are a positive constant ϵ = ϵ(n, β0 ) > 0 and sufficiently
large ℓ = ℓ(n, β0 ) such that for any conic Kähler-Einstein metric ω with angle
2πβ > 2πβ0 , we have ρω,ℓ ≥ ϵ.
In the following, we show the main steps in proving Theorem 4.2. As before, it suffices to prove the estimate ρωi ,ℓ ≥ ϵ uniformly for any sequence of
conic Kähler-Einstein metrics ωi ∈ K(M, D, t0 , β0 ) converging to a metric space
(M∞ , d∞ ) in the Gromov-Hausdorff topology, furthermore, we have
Ric(ωi ) = µi ωi + (1 − βi )[D]
11
with lim µi = µ∞ > 0 and lim βi = β∞ ≥ β0 .
First we establish the followings:
(F1) There is a uniform constant C = C(M, D, t0 , β0 ) such that for any ωi ∈
K(M, D, t0 , β0 ) and function f on M , we have the Sobolev inequality:
(∫
|f |
2n
n−1
ω
n
) n−1
n
(∫
)
≤ C
M
(|∇f | + |f | )ω
2
2
n
.
M
Applying this to (2.1) and (2.2) and the standard Moser iteration, we still have
the following estimates for ωi ∈ K(M, D, t0 , β0 ): There is a uniform constant
−ℓ
C0 such that for any σ ∈ H 0 (Mi , KM
) (ℓ > 0), we have
i
(
− 12
sup ||σ|| + ℓ
||∇σ||
)
(∫
≤ C0 ℓ
M
n
2
||σ|| ω
2
n
) 12
.
(4.5)
M
This can be proved by the same arguments in proving corresponding ones
for smooth Kähler-Einstein metrics.
(F2) There is a closed subset S ⊂ M∞ of Hausdorff codimension at least 2 such
that M∞ \S is a smooth Kähler manifold and d∞ is induced by a Kähler-Einstein
metric ω∞ outside S which satisfies
Ric(ω∞ ) = µ∞ ω∞ + (1 − β∞ )[D].
Moreover, ωi converges to ω∞ in the C ∞ -topology outside S.
(F3) For any ri 7→ 0, by taking a subsequence if necessary, (M∞ , ri−2 ω∞ , x)
converges to a tangent cone Cx at x satisfying:
1. Cx is a Kähler cone with vertex o;
2. Each Cx is regular outside a closed subcone Sx of complex codimension at
least 1. Such a Sx is the singular set of Cx ;
3. There is an natural Kähler Ricci-flat metric gx whose Kähler form ω∞ is
√
¯ 2 on Cx \Sx which is also a cone metric, where ρx denotes the distance
−1∂ ∂ρ
x
function from the vertex of Cx ;
4. For any 1 ≤ m ≤ n, define Sm to be the set of all points in S which have a
tangent cone of the form Cn−m × Cx′ , where Cx′ admits no line. Then
∪ Sm has
complex codimension at least m and S is the union of all Sm . Set S ′ = m≥2 Sm ,
then for any x ∈ S\S ′ , every tangent cone is of the form Cn−1 × Cβ , where Cβ
denotes the 2-dimensional flat cone of the angle 2πβ∞ ;
Both (F2) and (F3) can be proved by using the techniques developed in
[CCT95], but one needs new inputs in the proof which will appear in my joint
paper with Z.L. Zhang [TZ12].
−ℓ 5
¯ = 0 and
(F4) The L2 -estimate holds on M∞ : For any τ ∈ Λ0,1 (KM
) with ∂τ
∞
∫
−1
2 n
||τ || ω∞ < ∞, where || · || denotes a norm on KM∞ with curvature ω∞ ,
M∞
5 This
is understood as a section on the regular part of M∞ .
12
−ℓ
there is a σ ∈ KM
satisfying:
∞
∫
∂σ = τ and
M∞
n
||σ||2 ω∞
≤
1
ℓ+µ
∫
M∞
n
||τ ||2 ω∞
.
¯
This can be proved by applying the L -estimate for ∂-operator
to conic manifolds (M, D, ωi ) and taking the limit. This last step is similar to that in the
proof of Proposition 2.1.
Once we establish these, the remaining crucial ingredient is to have a version
of Lemma 2.4. Then we can proceed as in the proof of Theorem 1.5 except that
we need to choose different cut-off functions which correspond to the potential
of the Poincare metric on a punctured disc. The details will be presented later.
2
References
[CCT95] Cheeger, J., Colding, T. and Tian, G.: Constraints on singularities
under Ricci curvature bounds. C. R. Acad. Sci. Paris Sr. I, Math. 324
(1997), 645-649.
[CN10] Colding, T. and Naber, A.: Sharp Hlder continuity of tangent cones for
spaces with a lower Ricci curvature bound and applications. To appear
in Annals of Mathematics.
[DS12] Donaldson, S and Sun, S: Gromov-Hausdorff limits of Kḧler manifolds
and algebraic geometry. Preprint, arXiv:1206.2609.
[JMR11] Jeffres, T., Mazzeo, R. and Rubinstein, Y.: Khler-Einstein metrics
with edge singularities. arXiv:1105.5216.
[Ti87] Tian, G.: On Kähler-Einstein metrics on certain Kähler Manifolds with
C1 (M ) > 0. Invent. Math., 89 (1987), 225-246.
[Ti89] Tian, G.: On Calabi’s conjecture for complex surfaces with positive first
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[Ti90] Tian, G.: Kähler-Einstein on algebraic manifolds. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 587598,
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[Ti09] Tian, G.: Einstein metrics on Fano manifolds. ”Metric and Differential
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[Ti12] Tian, G.: Extremal Kähler metrics and K-stability. Preprint, June, 2012.
[TW11] Tian, G. and Wang, B.: On the structure of almost Einstein manifolds.
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[TZ12] Tian, G. and Zhang, Z.L.: In preparation.
13