Extremal Kähler metrics and K

Extremal Kähler metrics and K-stability
Gang Tian∗
Department of Mathematics
Beijing University and Princeton University
This is an expository paper dedicated to professor Calabi. In this paper,
I will discuss aspects of extremal Kähler metrics, an active research subject
which was initiated by E. Calabi. I will emphasis on the connection between
the existence of these metrics and the K-stability.
Contents
1 Preliminaries
1
2 Calabi-Futaki invariant
3
3 Kähler metrics of constant scalar curvature
6
4 The K-stability
8
5 Kähler-Einstein metrics on Fano manifolds
13
6 Conic Kähler-Einstein metrics
20
1
Preliminaries
A Kähler manifold is a complex manifold M which admits a Kähler metric. A
Kähler metric is a Riemannian metric which is Hermitian and makes parallel
the complex structure of M . It is determined by its Kähler form ω, in local
complex coordinates z1 , · · · , zn , the Kähler form is of the form
ω =
√
−1
n
∑
gij̄ dzi ∧ dz̄j ,
i,j=1
where n is the complex dimension of M and {gij̄ } is a positive Hermitian matrixvalued function such that dω = 0. We will simply use ω to denote both a metric
and its Kähler form.
∗ Supported
partially by a NSF grant
1
Recall that the Kähler class of ω is the cohomology class [ω] in H 2 (M, R)
represented by ω. It follows from the Hodge theory that if ω ′ is another Kähler
metric with [ω ′ ] = [ω], then there is a smooth function φ on M such that
√
ω ′ = ω + −1∂∂φ.
We will often denote the right side by ωφ . Thus, the space K[ω] of Kähler metrics
with the same Kähler class [ω] can be identified with
∫
√
φ ω n = 0, ω + −1∂∂φ > 0}.
K(M, ω) = {φ ∈ C ∞ (M, R) |
M
In [Cal82], E. Calabi proposed studying extremal Kähler metrics which are
those Kähler metrics which minimize the L2 -norm of the curvature tensor within
a given Kähler class. Using the Chern-Weil forms, he showed that this is equivalent to minimizing the L2 -norm of the scalar curvature. He further showed
that if ω is an extremal Kähler metric, then the scalar curvature s = s(ω) is a
holomorphic potential, that is, there is a holomorphic vector field X such that
iX ω =
√
∂s
∂
¯
−1∂s(ω),
or equivalently X i = g ij̄
, where X = X i
.
∂ z̄j
∂zi
In particular, Kähler metrics of constant scalar curvature form a special class
of extremal Kähler metrics. They are critical metrics of the K-energy which
was first introduced by T. Mabuchi [Ma86]: Fix a Kähler metric ω, for any
φ ∈ K(M, ω), define
∫ 1∫
Tω (φ) = −
φ̇t (s(ωφt ) − s) ωφnt dt,
(1.1)
0
M
where φt is a path in K(M, ω) with φ0 = 0 and φ1 = φ and φ̇t is the derivative
of φt in t. T. Mabuchi proved that it is independent of the choice of the path
φt , so it is a well-defined functional on K(M, ω).
If ω is of constant scalar curvature and π c1 (M ) = λ [ω], where c1 (M ) is the
first Chern class of M , then ω is actually Kähler-Einstein, i.e.,
Ric(ω) = λω,
(1.2)
where Ric(ω) denotes the Ricci curvature, in local coordinates z1 , · · · , zn ,
√
(1.3)
Ric(ω) = − −1∂∂ log det(gij̄ ).
The study of Kähler-Einstein metrics was initiated by E. Calabi in early 50’s,
in particular, he made the famous conjecture: Any (1,1)-form representing the
first Chern class is the Ricci curvature of a unique Kähler metric in a given
Kähler class. The uniqueness was already shown by E. Calabi in 50’s as an
application of the Maximum Principle. The Calabi conjecture was solved by
Yau in [Ya78]. In particular, it implies that if c1 (M ) = 0, then M admits a
Ricci-flat Kähler metric, now referred as a Calabi-Yau metric. E. Calabi also
2
proposed studying Kähler-Einstein metrics with non-vanishing scalar curvature.
Clearly, the necessary condition is that c1 (M ) = λ[ω] for some Kähler class and
real number λ. Without loss of generality, we may assume that λ = −1, 0, 1.
The case λ = 0 is taken care of in Yau’s solution of the Calabi conjecture. If
λ = −1 or equivalently c1 (M ) < 0, there is a unique Kähler-Einstein metric
satisfying (1.2) due to Aubin and Yau in 1976.
In case λ = 1, the uniqueness was proved by Bando and Mabuchi in [BM86]:
For any compact Kähler manifold M with c1 (M ) > 0, there is at most one
metric satisfying (1.2) up to automorphisms. The existence part for λ = 1 is
more tricky. If n = 2, M is a Del-Pezzo surface. I proved in [Ti90] that such an
M admits a Kähler-Eisntein metric if and only if the Lie algebra of holomorphic
vector fields on M is reductive. It is a driving force in current Kähler geometry
to study the existence of Kähler-Einstein metrics with positive scalar curvature
in higher dimensions. This will be one of main topics we will discuss in this
expository paper.
For general extremal Kähler metrics, we still have the uniqueness. More
precisely, in [Ch00], Chen proved the uniqueness theorem for Kähler metrics
of constant scalar curvature on any compact Kähler manifolds with negative
or vanishing first Chern class; In [ChTi], Chen and I proved that given any
compact Kähler manifold (M, ω), there is at most one extremal Kähler metric
ω ′ with [ω ′ ] = [ω] modulo automorphisms. When the automorphism group of
(M, ω) is discrete and [ω] is rational, this was also proved by S. Donaldson by
a different method. Also see [ChSo10] for alternative proof in special cases.
The existence problem on Kähler metrics of constant scalar curvature is
largely open in general. We have many examples of such metrics due to symmetry or the gluing method. There are holomorphic obstructions to the existence
which arise from holomorphic vector fields. We will discuss it in the next section.
It is now believed that the K-stability be the right condition for the existence of
Kähler metrics of constant scalar curvature. The large part of this expository
paper will be devoted to discussing various notions of the K-stability and its
relation to extremal Kähler metrics.
2
Calabi-Futaki invariant
In this section, we discuss a holomorphic obstruction found first by Futaki to
the existence of Kähler-Einstein metrics and generalized by E. Calabi to the
existence of general Kähler metrics of constant scalar curvature.
Definition 2.1. The Kähler cone Ka(M ) is the set of all cohomology classes
Ω ∈ H 2 (M, R) ∩ H 1,1 (M, C) which is the Kähler class of certain Kähler metric.
Let η(M ) be the space of holomorphic vector fields on M . The Calabi-Futaki
invariant is a character fΩ of η(M ) for each Ω ∈ Ka(M ) which is defined by
∫
fΩ (v) =
v(hω ) ω n , v ∈ η(M ),
(2.1)
M
3
where hω is determined by the equations
∫
s(ω) − sω = ∆ω hω ,
( hω
)
e − 1 ω n = 0,
M
where sω denotes the average of scalar curvature s(ω) and ∆ω is the Laplacian
operator of the metric ω. If c1 (M ) = λ[ω] for some λ > 0, Futaki proved in
[Fut83] that fΩ (v) is actually independent of the choice of ω. Later in [Cal85], E.
Calabi observed that Futaki’s arguments can be extended to any Kähler classes.
Therefore, (2.1) defines an invariant, often referred as the Calabi-Futaki invariant. Consequently, if M admits a Kähler metric of constant scalar curvature
within the Kähler class Ω, then fΩ ≡ 0. There are many examples of Kähler
manifolds with non-vanishing Futaki invariant, so there do not exist Kähler
metrics of constant scalar curvature on such manifolds. Furthermore, using
the independence of fΩ on metrics, one can show that fΩ ([v1 , v2 ]) = 0 for any
v1 , v2 ∈ η(M ), so it is a character of η(M ).
In the following, we will put the Calabi-Futaki invariant on a more conceptual
frame, namely, we will express it in terms of secondary classes: The Bott-Chern
classes [BC65]. First we observe the following:
Proposition 2.2. Let E be a Hermitian bundle with metric h over M and let
ϕ be a symmetric polynomial of degree n + 1. For any v ∈ η(M ), set
∫
F (v; ϕ) =
ϕ(θv + R(h), ..., θv + R(h)),
M
where θv ∈ End(E) and R(h) denotes the curvature form of h. Then F (v; ϕ) is
¯ v = −iv R(h).
independent of h whenever ∂θ
Remark 2.3. The integrand in F (v; ϕ) corresponds to one of the Bott-Chern
classes [BC65] along the holomorphic field v.
The Calabi-Futaki invariant can be expressed in terms of such F (v; ϕ)’s. To
see it, we first note that
sω =
nc1 (M ) · Ωn−1
.
Ωn
¯ v ω) = 0. Then the Hodge
Since ω is closed and v is holomorphic, we have ∂(i
Theory gives a smooth function θv and a harmonic 1-form α such that
¯ v.
iv ω = α − ∂θ
A direct computation shows
∫
hω ∆ω θv ω n .
fΩ (v) =
M
4
(2.2)
By a direct computation, we can shpw
fΩ (v) =
n
∑
j=0
(−1)j
j!(n − j)!
∫
(
(−∆ω θv + Ric(ω) + (n − 2j)(θv + ω))n+1
M
−(∆ω θv − Ric(ω) + (n − 2j)(θv + ω))n+1
n+1
∑ (n + 1 − 2j)n+1 ∫
(θv + ω)n+1
− sω
j!(n
+
1
−
j)!
M
j=0
)
¯ v = −iv ω and ∂∆
¯ ω θv = iv Ric(ω). It follows from
On the other hand, ∂θ
these and Proposition 2.2 that the Calabi-Futaki invariant fΩ is independent of
the particular representative ω chosen in [ω].
Given a Kähler metric ω, we can identify Tz M/Tz Z ≃ NM |Zλ , where NM |Zλ
is the normal bundle to Tz Zλ with respect to the metric ω.
Given any Kähler class Ω ∈ Ka(M ) we will define a “trace” trΩ (X) : {Zλ } →
C which is only well-defined up to addition of constants. Fixing ω with [ω] = Ω,
¯ v and we set trΩ (v)(Zλ ) = θv (Zλ ). Because
we may have that iv ω = −∂θ
¯
v|Zλ = 0, we see ∂θv = 0 and hence θv |Zλ is constant. For a different metric
¯ so θ′ = θv − v(ϕ) + c where c
ω ′ in the same class Ω, we have ω ′ = ω + ∂ ∂ϕ
v
is a constant independent of λ. It follows that the trace is well defined up to
addition of constants.
In the special case that Ω = c1 (M ), ω is the curvature of a hermitian metric
on the line bundle L = Λn T 1,0 M and we have the induced vector field v ∗ :
Λn T 1,0 → Λn T 1,0 which is the canonical lifting of v.
At z ∈ Zλ , Lz = z × C where we denote the coordinate on C by ξ so that
∂
the vector field v ∗ is given by v ∗ = a ∂ξ
where a = tr(Dv)(z). So in this case
trc1 (M ) (v)(z) = tr(Dv)(z) =
∑ ∂v i
∂zi
(z).
Theorem 2.4. For any (Ω, v) ∈ Ka(M ) × η(M ), we have
fΩ (v) =
∑∫
λ∈Λ
Zλ
[tr(Lλ (v)) + c1 (M ) −
n
n+1
sω (trΩ (v) + Ω)](trΩ (v) + Ω)n
det(Lλ (v) +
√
−1
2π Kλ )
.
Here Lλ (v) = (∇v)⊥ |M |Zλ and Kλ is the curvature form of the induced metric
on N |M |Zλ by ωg .
If Ω = c1 (M ) then this theorem is due to Futaki and reads
∫
1 ∑
tr(Lλ (v) + c1 (M ))n+1
√
fM (c1 (M ), v) =
.
−1
n+1
Zλ det(Lλ (v) +
Kλ )
λ∈Λ
2π
The Calabi-Futaki invariant can be generalized to singular varieties. In
[DiTi92], it is shown that one can define the invariant for normal Q-Fano varieties. In fact, the arguments can be applied to the general cases. Let (X, L)
5
be a polarized normal variety embedded in some projective space CP N and
kL = O(1)|M for some k > 0, where O(1) is the hyperplane bundle on CP N .
The standard metric on O(1) induces a Hermitian metric h on L and a metric ω
on X which is simply the curvature of h. Let η(X) be the algebra of the group
of automorphisms of X. Then for any v ∈ η(M ), there is an embedding such
that v is the restriction of a holomorphic vector field on the ambient projective
space. For simplicity, we may assume that this space coincides with CP N . The
arguments in [DiTi92] show that the integral in (2.1) still makes sense, moreover, its value is independent of the choices of embeddings, so it gives rise to
a generalized Calabi-Futaki invariant fc1 (L) : η(X) 7→ C. In principle, such a
generalized invariant can be defined for any polarized varieties in a similar and
analytic way, e.g., as a derivative of the K-energy (cf. [PT04] and its references).
In [Do02], Donaldson gave an algebraic definition of the Calabi-Futaki invariant which works for any polarized varieties. Let (V, L) be a n-dimensional
polarized variety and ρ : C∗ 7→ Aut(V, L) be a C∗ =action, where Aut(V, L) is
the group of automorphisms of V which can be lifted to L. Then the dimension
h ∗ 0(V, Lℓ ) and the weight w(V, L) of the C∗ -action induced on Λtop H 0 (V, L)
have the following asymptotic expansions as ℓ ≫ 0:
h0 (V, Lℓ )
w(V, Lℓ )
=
=
a0 ℓn + a1 ℓn−1 + O(ℓn−2 )
b0 ℓn+1 + b1 ℓn + O(ℓn−1 ).
(2.3)
(2.4)
The Donaldson’s version of the Futaki invariant is defined as
a1 b0 − b1 a0
.
a20
F (V, L; ρ) =
(2.5)
It is shown in [Do02] that F (V, L, ρ) is equal to the original Futaki invariant
(possibly modulo multiplication by a universal positive constant) whenever V
is smooth. In fact, the Donaldson-Futaki invariant F (V, L, ρ) coincides with
the generalized Calabi-Futaki invariant as Ding-Tian defined whenever V is a
normal variety as shown in [PT04] by using the CM line bundles.
3
Kähler metrics of constant scalar curvature
In this section, we collect some facts on Kähler metrics of constant scalar curvature. First we give an analytic criterion for their existence. Note that those
metrics are critical points of the K-energy Tω defined in (1.1).
Define
∫
√
1
φ(ω n − (ω + −1∂∂φ)n ),
(3.1)
Iω (φ) =
V M
∫
where V = M ω n . Also define
∫
Jω (φ) =
0
1
n−1
1 ∑ i+1
Iω (tφ)
dt =
t
V i=0 n + 1
√
∫
−1
∂φ ∧ ∂φ ∧ ω i ∧ ωφn−i−1 , (3.2)
2
M
6
where ωφ = ω +
√
−1∂∂φ. Notice that for any φ ∈ K(M, ω), we have
Iω (φ) ≥ 0, Jω (φ) ≥ 0, Iω (φ) − Jω (φ) ≥ 0
and
1
Jω (φ) ≤ Iω (φ) − Jω (φ) ≤ n Jω (φ).
n
Definition 3.1. We say that a Kähler manifold (M, ω) is analytically stable if
Tω is proper, that is, if there is an increasing function f (t) ≥ −c for some c ≥ 0,
where t ∈ (−∞, ∞), such that limt→∞ f (t) = ∞ and for any φ ∈ K(M, ω), we
have1
Tω (φ) ≥ f (Jω (φ)).
(3.3)
Some time, (M, ω) is said to be analytically semi-stable if Tω is bounded from
below.
If ω ′ = ωψ is another Kähler metric, then one can show
Tω (φ) = Tω′ (φ − ψ) + Tω (ψ).
(3.4)
Hence, the analytic stability depends only on the Kähler class of ω.
It was conjectured in [Ti98] that a compact Kähler manifold (M, ω) with
trivial Aut0 (M, [ω]) admits a Kähler metric of constant scalar curvature in the
Kähler class [ω] if and only if the polarized manifold (M, Ω) is analytically stable.
It is indeed true in the case of Kähler-Einstein metrics, that is, c1 (M ) = λ[ω]. If
λ ≤ 0, it follows from the Calabi-Yau theorem and the Aubin-Yau theorem and
the observation that the K-energy Tω is automatically proper in these cases. If
λ > 0, i.e., M is a Fano manifold, it was proved in [Ti97].
Theorem 3.2. Assume that c1 (M ) > 0 and η(M ) = {0}. Then M admits a
Kähler-Einstein metric if and only if (M, c1 (M )) is analytically stable.
Not much is known for general Kähler metrics with constant scalar curvature.
There are progresses on its version of this conjecture for toric surfaces (see
[Do02], [ZZ08], [ChLiSh] et al). However, we have
Theorem 3.3. If M admits a Kähler metric of constant scalar curvature and
with Kähler class [ω], then the K-energy Tω is bounded from below on K(M, ω),
i.e., (M, Ω) is analytically semi-stable.
This theorem was proved by Chen-Tian in [ChTi] in general cases and independently proved by S. Donaldson in [Do05] for projective manifolds without
non-zero holomorphic vector fields. In [Li09], Chi Li proved the above theorem
for projective manifolds. In the Fano case, this theorem was proved by BandoMabuchi [BM86] using the continuity method. This theorem was also extended
to the general cases of extremal Kähler metrics and Kähler-Ricci solitons (see
[ChTi], [TZ00] etc.).
1 One
can also use I − J instead of J in the definition.
7
Let us end up this section by showing a direct connection between the Kenergy and the Calabi-Futaki invariant. This connection was one of inspirations
for me to introduce the K-stability in [Ti97].
For any σ ∈ Aut(M, [ω]) and φ ∈ K(M, ω), there is a unique φσ such that
∫
√
( hω −φσ
)
¯ σ and
σ ∗ ωφ = ω + −1∂ ∂φ
e
− 1 ω n = 0.
M
Assume that v ∈ η(M ) is a holomorphic vector field such that there is a unique
θv satisfying:
∫
√
¯ v and
iv ω = −1∂θ
θv ehω ω n = 0.
M
It follows from these and a direct computation
∆ω θv + θv + v(hω ) = 0.
(3.5)
If σ(t) is a one-parameter subgroup generated by the real part Re(v) of v,
then φ̇σ(t) = θv (t), where θv (t) is the corresponding potential θv of v when ω is
replaced by ωφ for φ = φσ(t) . It follows
dTω (φσ(t) )
= − Re
dt
(
1
V
∫
v(hσ∗ ωφ ) σ
∗
)
ωφn
.
(3.6)
M
The last integral is simply the Calabi-Futaki invariant. Therefore, we have
Proposition 3.4. If (M, ω) is analytically semi-stable, then the Calabi-Futaki
invariant f[ω] vanishes.
4
The K-stability
In this section, we discuss the K-stability. We will first present the definition
from [Ti97] by using the generalized Calabi-Futaki invariants, then we show
a more algebraic definition due to Donaldson [Do02]. For simplicity, we will
assume that the automorphism group Aut(M, [ω]) is discrete. Note that most
Kähler manifolds satisfy this assumption.
To motivate the introduction of the K-stability, we recall a result proved in
my 1988 thesis. Let (M, L) be a polarized manifold. By the Kodaira embedding theorem, for ℓ sufficiently large, a basis of H 0 (M, Lℓ ) gives an embedding
ϕℓ : M 7→ CP N , where N = dimC H 0 (M, Lℓ ) − 1. Any other basis gives an
embedding of the form σ · ϕℓ , where σ ∈ G = SL(N + 1, C). We fix such an
embedding and consider the action of G on M .
Fix a Hermitian metric || · || on L such that its curvature form ω is a Kähler
metric. Then for any σ ∈ G, there is a unique function φσ ∈ K(M, ω) such that
√
1 ∗ ∗
¯ σ,
ϕℓ σ (ωF S ) = ω + −1∂ ∂φ
ℓ
8
(4.1)
where ωF S is the Fubini-Study metric on CP N . Denote by Kℓ (M, ω) the set of
all such φσ . It was proved in [Ti88] (also see [Ti90]) that K(M, ω) is the closure
of the union of Kℓ (M, ω) for all sufficiently large ℓ in the C 4 -topology. Later,
it was shown the same holds in the C ∞ -topology due to the works of Ruan,
Zelditch and Catlin ([Ru97], [Ze], [Cat]). One can make the involved convergence
more precise. Given any φ ∈ K(M, ω), one can associate a Hermitian norm
|| · ||φ = e−φ || · || on each Lℓ whose curvature is the metric ℓωφ . Let {si }0≤i≤N
be an orthonormal basis of H 0 (M, Lℓ ) with respect to the induced inner product,
we set
N
∑
ρℓ (ωφ )(x) =
||si ||2φ (x),
x ∈ M.
i=0
Then we have
Theorem 4.1. ([Ti88],[Cat], [Ze], [Ru97], [Lu00]) There are smooth functions
ai (ω) on M such that for any r, k > 0,
||ρℓ (ω) −
k
∑
ai (ω)ℓn−i ||C r (M ) ≤ Cr,k,ω ℓn−k−1
i=0
for some constant Cr,k,ω . Furthermore, a0 (ω) = 1 and a1 (ω) = s(ω)/2, the
half of the scalar curvature of ω;
As discussed in last section, we expect that the existence of Kähler metrics with constant scalar curvature and Kähler class c1 (L) is equivalent to the
properness of the K-energy Tω with [ω] = c1 (L). In view of above theorem, we
believe that the properness of the K-energy on K(M, ω) follows from the properness of the K-energy restricted to Kℓ (M, ω) for all sufficiently large ℓ. On the
other hand, the basic idea in [Ti97] is: The K-stability for the embedding
of M ,→ CP N by using H 0 (M, Lℓ ) is equivalent to the properness of the
K-energy restricted to Kℓ (M, ω). Now let me show in more details.
For any algebraic subgroup G0 = {σt }t∈C∗ of SL(N +1, C), there is a unique
limiting cycle
M0 = lim σt (M ) ⊂ CP N .
t→0
Let v be the holomorphic vector field whose real part generates the action by
σ(e−s ), that is,
dσ(e−s )
= Re(v)(σ(e−s )).
ds
It follows from [DiTi92]2 that if M0 is a normal variety, then
lim
s→∞
)
d (
Tω (φσ(e−s ) ) = Re(fM0 ,Ω (v)),
ds
(4.2)
where Ω = 1ℓ [ωF S ]|M0 . We will denote by w(M, L, G0 ) the limit in (4.2) and call
it the weight of G0 associated to (M, L) (abbreviated as w(G0 ) if no confusion).
2 Only the Fano case was stated in [DiTi92], but the arguments clearly applied to the
general cases.
9
The K-stability was first introduced in [Ti97] by using generalized CalabiFutaki invariants. In fact, the definition was stated only for Fano manifolds
though it clearly works for any polarized manifolds (M, L). A Fano manifold M
has positive first Chern class, consequently, it has a canonical polarization which
is its anti-canonical bundle. We will first discuss the case for Fano manifolds as
in [Ti97] and then the general cases. We always denote by G0 an one-parameter
subgroup of SL(N + 1) and by M0 the corresponding limit cycle.
Definition 4.2. Let M be a Fano manifold. We say that M is K-semistable
−ℓ
with respect to KM
if w(G0 ) ≥ 0 for any G0 ⊂ SL(N + 1) such that the
corresponding M0 is a normal variety. We say that M is K-stable with respect
−ℓ
to KM
if it is K-semistable and w(G0 ) > 0 for any G0 ⊂ SL(N + 1) such that
the corresponding M0 is a normal variety and not biholomorphic to M .
The following is proved in [Ti97]
Theorem 4.3. Let M be a Fano manifold without non-trivial holomorphic vector fields and which admits a Kähler-Einstein metric. Then M is K-stable with
−ℓ
respect to any very ample KM
.
The reason for imposing the normality condition on M0 is because of the
compactness theory for Riemannian manifolds with Ricci curvature bounded
from below. In establishing the existence of Kähler-Einstein metrics with positive scalar curvature, we can use the continuity method, then we are led to
studying the limit of a sequence of Kähler metrics satisfying a complex MongeAmpere equation, in particular, their Ricci curvature is bounded from below
by a positive constant and their volume is fixed. For those Kähler metrics,
we expect that the limit has essential singularity only along a closed subset of
Hausdorff codimension at least 4. Indeed, if this sequence further has bounded
Ricci curvature, the limit is C 1,α -smooth outside a closed subset of codimension
4 (see [CCT95]).
Now we extend the definition of the K-stability to general polarized manifolds. As above, (M, L) is a polarized manifold. Assume that M ⊂ CP N and
the hyperplane bundle O(1) of CP N restricts to Lℓ on M . For any algebraic
subgroup G0 = {σt }t∈C∗ of SL(N + 1, C), there is a unique limiting cycle
M0 = lim σt (M ) ⊂ CP N .
t→0
First we observe that the limit in (4.2) exists even if M0 is not normal. We still
denote by w(M, L, G0 ), the weight of G0 associated to (M, L) and abbreviated
as w(G0 ) if no confusion. Actually, it follows from [Ti97] and [PT04] 3 that
Tω (φσ(t) ) = (F (M0 , Lℓ , G0 ) + a) log
1
+ O(1),
t
(4.3)
where F (M0 , Lℓ , G0 ) is the Donaldson-Futaki invariant and a ∈ Q is negative if
and only if the limit cycle M0 has generically non-reduced fibres.4
3 This
was pointed out to us by Chi Li.
is shown in [PT04] that the Donaldson-Futaki invariant coincides with the weight on
the CM line bundle introduced in [Ti97].
4 It
10
Definition 4.4. Let (M, L) be a polarized Kähler manifold and M ⊂ CP N by
sections of Lℓ . We say that M is K-semistable with respect to Lℓ if w(G0 ) ≥ 0
for G0 ⊂ SL(N +1) such that the corresponding M0 has no multiple components.
We say that M is K-stable with respect to L−ℓ if it is K-semistable and w(G0 ) >
0 for G0 ⊂ SL(N + 1) such that the corresponding M0 is normal and is not
biholomorphic to M .
The reason for our imposing weaker condition on M0 in general cases is still
because of the compactness theory for metrics satisfying certain curvature conditions. In general cases, the compactness fails only along a subset of Hausdorff
codimension at least 2.
In [Do02], Donaldson gave an algebraic version of the K-stability. First
we recall that a test configuration of a polarized manifold (M, L) consists of a
scheme M endowed with a C∗ -action that linearizes on a line bundle L over
M, and a flat C∗ -equivariant map π : M 7→ C such that L|π−1 (0) is ample on
π −1 (0) and we have (π −1 (1), L|π−1 (1) ) ∼
= (M ; Lr ) for some r > 0. When (M, L)
∗
∗
has a C -action ρ : C 7→ Aut(M ), a test configuration where M = M × C and
C∗ acts on M diagonally through ρ is called product configuration.
Now we can state Donaldson’s version of the K-stability.
Definition 4.5. A polarized Kähler manifold (M, L) is K-semistable if for each
test configuration π : (M, L) 7→ C for (M, L) the Donaldson-Futaki invariant
of the induced action on the central fiber (f −1 (0), L|f −1 (0) ) is non-negative. We
say (M, L) is K-stable if it is K-semistable and the Donaldson-Futaki invariant
for any test configuration with normal central fiber 5 strictly positive unless we
have a product configuration.
Clearly, Donaldson’s version of the K-stability fits nicely with the usual way
of studying algebraic manifolds. In view of the identification between the original Calabi-Futaki invariant and the Donaldson-Futaki invariant for reduced
varieties mentioned before, one can see that the basic difference between Donaldson’s version and the earlier version is the condition on the central cycle M0 .
However, I expect that the above two versions of the K-stability are equivalent.
Indeed, it follows from a recent work by Arezzo-La Nave-Della Vedova [ALV09]
that the two versions of the K-semistability are equivalent.
Recently, Li and Xu proved the following which solves a conjecture of mine.
Theorem 4.6. A Fano manifold M is destablized by a test configuration, then
M is also destablized by a test configuration with normal central fiber.
The importance of the K-stability is partly shown in the following result of
J. Stoppa [Sto07].
Theorem 4.7. Let (M, L) be a polarized Kähler manifold with trivial aut(M, L).
Then M admits a Kähler metric with constant scalar curvature with Kähler class
c1 (L) only if (M, L) is K-stable.
5 This assumption on normality was not in the original definition of Donaldson. An example
of Li-Xu in [LX11] shows that it is necessary.
11
Its proof is based on Arezzo-Pacard’s work [ArPa05] and the works on the
K-semi-stability ([ChTi], [Do05]). In [ChTi] and [Do05], the authors prove independently that M admits a Kähler metric with constant scalar curvature with
Kähler class c1 (L) only if (M, L) is K-semistable. Recently, Stoppa’s result has
been extended by T. Mabuchi to the general case.
The following folklore conjecture also illustrates the importance of the Kstability.
Conjecture 4.8. Let (M, L) be a polarized Kähler manifold. For simplicity, assume that Aut(M, L) is discrete. Then M admits a Kähler metric with constant
scalar curvature and Kähler class c1 (L) if and only if (M, L) is asymptotically
K-stable.6
Finally, let us briefly discuss Sean Paul’s formulation of the K-stability. It
is in terms of hyperdiscriminants. For any projective manifold M ⊂ CP N , the
rth-hyperdiscriminant ∆M (r) is defined as follows (0 ≤ r ≤ n): First, using the
Segre embedding ιr : CP N × CP r ,→ CP Nr , where Nr + 1 = (r + 1)(N + 1), we
can regard M × CP r ⊂ CP Nr , then all the hyperplanes of CP Nr which contain
a tangent space of M × CP r form a hypersurface in the dual projective space
CP Nr ,∨ . The rth-hyperdisciminant ∆M (r) is simply a defining polynomial of
this hypersurface. Note that ∆M (n) coincides with the Chow form RM .
Let H ⊂ SL(N + 1, C) be a maximal algebraic torus and MZ (H) be its
character lattice (∼
= ZN ). Set
N (r) ⊂ MR (H) := MZ (H) ⊗Z R
to be the weighted polytope of ∆M (r) through the induced representation of G
in the space of homogeneous polynomials on CP Nr +1,∨ of degree=deg ∆M (r).
It follows from a result of Sean Paul that for any one-parameter subgroup
G0 ⊂ H, the weight F (M0 , Lℓ , G0 ) + a in (4.3) is equal to
deg(∆X (n))
min
x∈N (n−1)
lλ (x) − deg(∆X (n − 1)) min lλ (x)
N (n)
This implies N (∆M (n)) is contained in N (∆M (n − 1)) up to scaling if the
K-energy Tω is bounded from below. If K-energy is proper, then we can get
stronger inclusion relation. The following definition is due to S. Paul [Paul08].
Definition 4.9 (Polytope K-stability). We say M is polytope K-semistable if
for any maximal torus H ⊂ SL(N + 1, C), we have
deg(∆M (n − 1))N (n) ⊆ deg(∆M (n))N (n − 1).
We say M is polytope K-stable if for any maximal torus H and all m > 0
sufficiently large, we have
deg(∆M (n − 1))[(m − 1)N (n) + deg(RM )SN ] ⊆ m deg(RM )N (∆M (n − 1))
SN is the standard N-simplex in RN , and the addition on the left side denotes
Minkowski summation of polyhedra.
6 This
conjecture is often referred to the Yau-Tian-Donaldson conjecture.
12
Using known results on the K-energy, S. Paul proved
Theorem 4.10. If M admits a Kähler metric ω of constant scalar curvature
and with [ω] = λc1 (L) for some λ > 0 and line bundle L, then M is polytope
K-semistable with respect to any Lm . Furthermore, if M has a Kähler-Einstein
metric and has no non-trivial holomorphic vector fields, then M is polytope
K-stable with respect to a natural polarization.
This theorem has very interesting geometric consequences. For instance, if
C ⊂ CP N is a smooth curve of genus g, then ∆C (n − 1) = ∆C = C ∨ is simply
the dual variety of the curve C. Then by the Riemann uniformization theorem,
C has a Kähler-Einstein metric, so M is polytope K-semistable, in particular,
(
)
g−1
1+
N (RM ) ⊆ N (∆M )
d
Now assume C ⊂ CP N with L = O(1)|C and of degree d, then by Mumford, if
d ≥ 2g ≥ 0 then C is Chow-Mumford semistable. This is equivalent to saying
that N (RC ) contains 0. By the above relation, N (∆C ) also contains 0, so we
deduce the following geometric consequence: If C ,→ CP N be a smooth linearly
normal algebraic curve such that deg(C) ≥ 2g. Then ∆C is stable for the action
SL(N + 1, C). Note that C ∨ usually has singularities and most singular curves
are not semistable.
5
Kähler-Einstein metrics on Fano manifolds
In this section, I discuss a program I started in the late 80’s. The two main
ingredients in this program is the K-stability and a partial C 0 -estimate. We
have seen the K-stability in previous sections, so we will focus on the partial
C 0 -estimate. Our discussion here is an expansion of what we did in [Ti09]. A
successful example of this program is my solution for the existence of KählerEinstein metrics on Del-Pezzo surfaces (see [Ti89]).
First we recall Theorem 3.2: If c1 (M ) > 0 and η(M ) = {0}, then M admits
a Kähler-Einstein metric if and only if (M, c1 (M )) is analytically stable (cf.
Theorem 3.2). This means that in order to establish the existence of KählerEinstein metrics on a Fano manifold M , we basically need to prove that the
K-energy is proper as in Definition 3.1. It follows from [Ti97] that the K−ℓ
stability for the embedding of M ,→ CP N by using H 0 (M, KM
) is equivalent
to the properness of the K-energy restricted to Kℓ (M, ω), where ω represents
c1 (M ). The bridge between the properness of the K-energy on K(M, ω) and
restricted to Kℓ (M, ω) is given by the partial C 0 -estimate I conjectured in late
80’s (see [Ti89] and [Ti90]). Now let me state this conjecture and discuss what
we have known about it.
Consider the set
K(M, t0 ) = { ω̃ | [ω̃] = c1 (M ), Ric(ω̃) ≥ t0 ω̃ }.
13
This is clearly a subset of K(M, ω). For any ω̃ ∈ K(M, t0 ), choose a Hermitian
metric h̃ with ω̃ as its curvature form and any orthonormal basis {Si }0≤i≤N of
−ℓ
) with respect to an induced inner product. Put
each H 0 (M, KM
ρω̃,ℓ (x) =
N
∑
||Si ||2h̃ (x).
(5.1)
i=0
This is independent of the choice of h̃ and the orthonormal basis {Si }.
Conjecture 5.1. [Ti90] There are uniform constants ck = c(k, n) > 0 for k ≥ 1
and ℓi → ∞ such that for any ω̃ ∈ K(M, t0 ) and ℓ = ℓi for each i,
ρω̃,ℓ ≥ cℓ > 0.
(5.2)
Remark 5.2. There is a uniform upper bound on ρω̃,ℓ which depends only on
n and ℓ. This is proved in [Ti90]. The lower bound is more important and can
be regarded as an effective version of the very ampleness in algebraic geometry.
Remark 5.3. In fact, I expect the stronger version of Conjecture 5.1: There
are uniform constants ck = c(k, n) > 0 for k ≥ 1 and ℓ0 = ℓ0 (n) such that for
any ω̃ ∈ K(M, t0 ), and ℓ ≥ ℓ0 , ρω̃,ℓ ≥ cℓ .
Let us see how this conjecture is related to the existence of Kähler-Einstein
metrics on M . It is proved in [Ti97] that if a Fano manifold M admits a KählerEinstein metric and has no non-trivial holomorphic fields, then M is K-stable
with respect to K −ℓ for all sufficiently large ℓ. Now we assume that M is
K-stable with respect to K −ℓ for all sufficiently large ℓ, we will explain how
Conjecture 5.1 implies the existence of Kähler-Einstein metrics. An approach
to proving such an existence is to solve the following equations:
(ω +
√
¯ n = ehω −tφ ω n .
−1∂ ∂φ)
(5.3)
It is known that in order to establish the existence of Kähler-Einstein metrics
on M , we only need to establish the a priori C 0 -estimate for the solutions of
(5.3) for t ≥ t0 for some t0 > 0 which may depend on (M, ω) (cf. [Ti98]).
Denote by φ the solution of (5.3) for some t ≥ t0 . Since the K-energy Tω is
monotonically decreasing along (5.3) (cf. [Ti98]), we have a uniform bound
Tω (φ) ≤ C = C(ω).
(5.4)
It is known that ωt = ωφ has its Ricci curvature greater than and equal to t0 ,
so by Conjecture 5.1, we have an embedding σ : M ,→ CP N by an orthonormal
−ℓ
basis of H 0 (M, KM
) with respect to ωt for some large ℓ > 0, such that
√
1 ∗
¯
σ ωF S = ω + −1∂ ∂ψ,
||ψ − φ||C 0 ≤ C.
ℓ
Here C denotes a uniform constant. In fact,
(N
)
∑
1
2
ψ − φ = log
||Sa || ,
ℓ
a=0
14
(5.5)
where ||·|| is a Hermitian norm whose curvature is ωt and {Sa } is an orthonormal
basis with respect to ωt . It follows
Tω (ψ) ≤ C ′ = C ′ (ω).
(5.6)
−ℓ
Since the K-stability of (M, KM
) is equivalent to the properness of Tω restricted to Kℓ (M, ω), it follows from our assumption and (5.6) that ψ is uniformly bounded. Hence, φ stays uniformly bounded and consequently, there is
a Kähler-Einstein metric on M .
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, if ℓ is sufficiently large, for each i, there is an embedding
−ℓ
σi : M ,→ 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 5.1 and even its stronger version.
The following had come to my attention in early 90’s:
Conjecture 5.4. The Gromov-Hausdorff limit M∞ can be identified with the
complex limit M̄∞ .
This conjecture implies the partial C 0 -estimate. If ωi are Kähler-Einstein
metrics, one can also prove the converse is true (see [Ti09]). In fact, we conjecture that Conjecture 5.1 and Conjecture 5.4 are equivalent even in general
cases.
In the following, we consider the special case of Conjecture 5.1 for KählerEinstein metrics, i.e., those ω with Ric(ω) = ω. In this special case, we can apply
certain works of Cheeger-Colding-Tian [CCT95] which give more regularity on
the convergence of (M, ωi ) and (M∞ , d∞ ). More precisely, we have
Theorem 5.5. [CCT95] Let (Mi , ωi ) be a sequence of Kähler-Einstein manifolds with Ric(ωi ) = ωi and which converges to (M∞ , d∞ ) in the GromovHausdorff topology. Then 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.
Even though S remains to be understood, we can treat M∞ as a “good”
−ℓ
variety in many ways. By an element of H 0 (M∞ , KM
), we mean a holomorphic
∞
−ℓ
2
7
section σ of KM∞ on M∞ \S with finite L -norm Then one can prove that
−ℓ
the space H 0 (M∞ , KM
) is of finite dimension. Choose a Hermitian metric
∞
−1
h∞ of KM∞ outside S with ω∞ as its curvature. We may also choose hi for
−1
KM
such that hi converges to h∞ outside S. The 2-dimensional version of the
i
following proposition was proved in [Ti89] and the same proof works for higher
dimensions.
7 This
should be automatically true since S is of codimension at least 4.
15
Proposition 5.6. By taking a subsequence if necessary, for each ℓ, we have
−ℓ
−ℓ
) converges to H 0 (M∞ , KM
) as i tends to ∞ in the sense:
that H 0 (Mi , KM
i
∞
−ℓ
i
There are orthonormal bases {σa }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. First we observe (see [Ti90]):
Lemma 5.7.
8
For each i, we have the following identities:
∆ωi ||σai ||2 = ||∇σai ||2 − nℓ ||σai ||2 ;
∆ωi ||∇σai ||2 = ||∇2 σai ||2 − ((n + 2) ℓ − 1) ||∇σai ||2 .
It follows from this and the Moser iteration that we can bound each σai uniformly as well as the derivatives of σai outside the singular set of M∞ . Therefore,
by taking a subsequence if necessary, we can assume that σai converges to a σa∞
as i tends to ∞.
To complete the proof of last proposition, we only need to show that each
−ℓ
section σ̃∞ of KM
on M∞ is the limit of a sequence of sections σ̃i on Mi . This
∞
can be done by using Hömander’s L2 -estimate. For any ϵ > 0, choose a finite
cover {Brk (xk )} of the singular set S ⊂ M∞ satisfying:
1. r∑
k ≤ ϵ;
2. k rk2n−4 ≤ C < ∞;9
3. For any x ∈ M∞ , the number of balls Brk (xk ) containing x is uniformly
bounded, say C.
Let η be a smooth function on R such that η(r) = 0 for r ≤ 1 and η(r) = 1
for r ≥ 2. Put
)
(
∏
′
η(d(·, xk )/rk ) σ̃∞ .
σ∞ =
k
Then one can show
∫
M∞
¯ ′ ||2 ω n ≤ Cϵ2 .
||∂σ
∞
∞
For i sufficiently large, there is a diffeomorphism Φi : Mi \Ki 7→ M∞ \S satisfying:
(1) The measure of Ki is bounded by Cϵ4 ;
(2) The image of Φi contains the complement of ∪k Brk (xk ) in M∞ ;
¯ i || ≤ ϵ.
(3) ||∂Φ
−ℓ
Each Φi induces an isomorphism τi between the complex line bundles KM
i
−ℓ
¯ i || ≤ Cℓ ϵ, where Cℓ is independent of i and ϵ. Put
and KM
such
that
||
∂τ
∞
8A
similar lemma works for general ωi as this one does.
will always use C to denote a uniform constant in this proof, though its actual value
may vary in different places.
9 We
16
′
σi′ = τ −1 (σ∞
). Then
∫
¯ ′ ||2 ω n ≤ Cℓ ϵ2 .
||∂σ
i
i
Mi
¯
By applying Hömander’s L -estimate for ∂-operators,
we can have a C ∞ -section
−ℓ
ui of KMi satisfying:
∫
¯ i = −∂σ
¯ ′,
∂u
||ui ||2 ωin ≤ Cℓ ϵ2 .
i
2
Mi
−ℓ
It implies that σ̃i = σi′ +ui is a holomorphic section of KM
such that σ̃i is within
i
the Cϵ-distance of σ̃∞ for some uniform constant C. Then the proposition
follows easily.
Proposition 5.6 can be regarded as a metric version of the flatness for families
of projective varieties. Thus, in order to prove (5.2), we only need to show
(N
)
∑
∞ 2
inf
(5.7)
||σa ||ω∞ > 0,
x∈M∞
a=0
where σa∞ are given in Proposition 5.6. This is simply
inf ρω∞ ,ℓ (x) > 0.
x
This lower bound can be achieved by using the L2 -estimate and the structure
results on M∞ which follow from the regularity theory due to Cheeger-ColdingTian [CCT95], more precisely, we claim
Theorem 5.8. Conjecture 5.1 holds for K(M, 1).
As a corollary, we have
Corollary 5.9. Conjecture 5.4 holds for K(M, 1).
I will outline the proof of Theorem 5.8 and leave the details to [Ti12].
First observe that by using Lemma 5.7, one can show that ρωi ,ℓ are uniformly
continuous, so ρω∞ ,ℓ is continuous on M∞ . According to [CCT95], for any
x ∈ M∞ and λi 7→ +∞, by taking a subsequence if necessary, we have a tangent
cone Cx of (M∞ , ω∞ ) at x, where Cx is the limit limi→∞ (M∞ , λi ω∞ , x) in the
Gromov-Hausdorff topology, satisfying:
1. Cx is a Kähler cone;
2. Cx = Cn−m × Cx′ , where m ≥ 2 and Cx′ does not contain any geodesic line.
If we set
Sm = {x ∈ M∞ | Cx = Cn−m × Cx′ },
then
∪ by Cheeger-Colding, Sm is of complex codimension at least m, in particular,
m>2 Sm has complex codimension at least 3.
First we show why for any x ∈ S2 , ρω∞ ,ℓ (x) > 0 for sufficiently large ℓ. Since
x ∈ S2 , there is a sequence {ri } with limi→0 ri = 0 such that (M∞ , ri−2 ω∞ , x)
17
converge to a tangent cone Cx = Cn−2 ×C2 /Γ, where Γ ⊂ U (2) is a finite group.
Fix a small ϵ ∈ (0, 1) which will be determined later. Then there are xi ∈ Mi
and r > 0 satisfying:
1. limi→∞ xi = x;
2. There are diffeomorphisms ϕi : B3n−2 (0) × A(ϵ, 2)/Γ 7→ Ai ⊂ Mi , where
2
2
k
A(R′ , R) = BR
(0)\BR
′ (0) and BR (0) denotes the ball with radius R and centered at 0, such that
||r−2 ϕ∗i ωi − ωe ||C 8 (B n−2 (0)×A(ϵ,2)/Γ) ≤ ϵ,
3
(5.8)
where the norm is given with respect to the flat metric ωe ;
3. The distance between xi and ϕi (B1n−2 (0) × A(ϵ, 1)/Γ) is less than 2ϵ;
4. For any holomorphic function f on Ai bounded by a uniform constant C
which may depend on x, we have
|f | ≥ 2 on ϕi (B1n−2 (0) × ∂B12 (0)/Γ) ⇒ |f | ≥ 1 on ϕi (B1n−2 (0) × A(ϵ, 1)/Γ).
All except Property 4 are clear from the compactness theory. Property 4 can
be proved by the contradiction argument and the Hartog’s theorem on A(ϵ, 1).
Now we can apply the L2 -estimate to prove our claim. Let ℓ be a multiple of
|Γ|, then by the definition of Ai , for each i sufficiently large, there is a trivializing
−ℓ
section τi of KM
with ||τi || = 1 + o(1). Using the cut-off function as we did
i
in the proof of Proposition 5.6, we can find a C ∞ -section τ̃i satisfying:
n−2
(i) τ̃i coincides with τi on ϕi (B3/2
(0) × A(2ϵ, 3/2)/Γ);
∫
(ii) M ||∂¯τ̃i ||2 ωin ≤ Cr2n−2 .
Note that C, C ′ et al always denote uniform constants in this section. By
−ℓ
¯
¯ i = ∂¯τ̃i
the L2 -estimate for ∂-operator,
we get a section vi of KM
such that ∂v
i
and
∫
∫
1
2 n
||vi || ωi ≤
||∂¯τ̃i ||2 ωin ≤ Cr2n−2 ℓ−1 .
ℓ Mi
Mi
−ℓ
Then σi = τ̃i − vi is a holomorphic section of KM
. By (i),
i
3
on ϕi (B n−2
(0) × A(2ϵ, )/Γ),
3
2
2
then by applying the standard elliptic estimates to the first identity in Lemma
5.7, we can get
∫
sup
||vi ||2 ωin ≤ C ′ r2 ℓ−1 .
||vi ||2 ≤ Cr−2n
¯ i = 0
∂v
ϕi (B1n−2 (0)×A( 21 , 32 )/Γ)
Mi
Choosing ℓ = 4C ′ r−2 , we can show ||σi || ≥ 1/2 on ϕi (B1n−2 (0) × A( 21 , 23 )/Γ).
On the other hand, it is clear that ||σi || are uniformly bounded. It follows from
the above 4 that ||σi || ≥ 1/4 on ϕi (B1n−2 (0) × A(ϵ, 1)/Γ).
18
Now we apply Lemma 5.7 to σi and get
∫
∫
2
′′ −2n
2 n
′′ −2n
sup ||∇σi || ≤ C r
||∇σi || ωi = nC r
ℓ
Mi
Mi
||σi ||2 ωin ≤ C̃ℓ
Mi
Hence, if 100ϵ < (C ′ C̃)−1 , then ||σi ||(xi ) ≥ 1/8, so ρω∞ ,ℓ (x) > 0, in fact, it is
bounded from below by a uniform constant depending only on ℓ or r.
Thus by replacing ℓ by kℓ for a large k, we get an embedding Φi : Mi 7→ CP N
−ℓ
by using an orthonormal basis {σai } of H 0 (Mi , KM
) such that Φi converge to a
i
N
rational map Φ∞ : M∞ 99K CP which is a holomorphic embedding near x. It
follows that M∞ is a variety near x and ∪
the singular stratum S2 is a subvariety
near x. Since x is arbitrary in S2 , M∞ \ m>2 Sm is a variety with singular set
S2 10 .
The above arguments can be extended to higher strata Sm for m > 2. For
each x ∈ Sm (m > 2). Any tangent cone Cx of (M∞ , ω∞ ) is of the form
Cn−m × Cx′ . One can show that if we denote by Reg(Cx′ ) the regular part of
Cx′ , then the identity component of its holonomy group lies in SU (n). This is
a crucial observation and allows us to construct a parallel section of the k-th
tensor product of the canonical bundle of Reg(Cx′ ) for some uniform k. If Cx′ is
smooth outside the vertex 0, then it is clear that Cx′ \{0} has trivial canonical
bundle modulo a finite cover and we can proceed as above. In this case, for any
small ϵ ∈ (0, 1), we can still find xi ∈ Mi and r > 0 satisfying:
1. limi→∞ xi = x;
2. There are diffeomorphisms ϕi : B3n−m (0) × A(ϵ, 2) 7→ Ai ⊂ Mi , where
m
m
m
′
A(R′ , R) = BR
(0)\BR
′ (0) and BR (0) ⊂ Cx denotes the ball with radius R and
centered at the vertex 0, such that
||r−2 ϕ∗i ωi − ωc ||C 8 (B n−m (0)×A(ϵ,2)) ≤ ϵ,
3
(5.9)
where the norm is given with respect to the cone metric ωc ;
3. The distance between xi and ϕi (B1n−m (0) × A(ϵ, 1)) is less than 2ϵ;
4. For any holomorphic function f on Ai bounded by a uniform constant C
which may depend on x, we have
|f | ≥ 2 on ϕi (B1n−m (0) × ∂B12 (0)) ⇒ |f | ≥ 1 on ϕi (B1n−m (0) × A(ϵ, 1)).
Those can be established in a rather standard way, furthermore, Cx′ has a
l
parallel section τ for some KC
′ . Using these, we ca argue as above to show that
x
ρω∞ ,ℓ (x) > 0.
If Cx′ has non-isolated singularities, using what we have proved for smaller m
and doing induction on the dimension of the singular set, we can still show that
the identity component of its holonomy group of Reg(Cx′ ) lies in SU (n). Next
10 In
fact, one can show that S2 consists of quotient singularities.
19
we construct approximated holomorphic section τi by using parallel sections
we have on Reg(Cx′ ) and proceed in a way similar to the above. Then we can
complete the proof of Theorem 5.8, and consequently, Corollary 5.9. Since the
arguments are rather lengthy and technical, we will leave details to [Ti12].
6
Conic Kähler-Einstein metrics
The theory of smooth Kähler-Einstein metrics can be generalized to the metrics
with conic angle along a divisor.
Let D ⊂ M be a smooth divisor in a compact Kähler manifold. 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 at D to the model conic metric
gβ = |z1 |2β−2 |dz1 |2 +
n
∑
|dzj |2 ,
j=2
where z1 , z2 , · · · , zn are holomorphic coordinates such that D = {z1 = 0} locally.
A conic Kähler-Einstein metric is a conic Kähler metric which are also Einstein.
The problem on the existence of conic Kähler-Einstein metrics was first
raised in [Ti94], even for more general cases where D may have singularities
which are simple normal crossings. It was also anticipated that the complete
Ricci-flat Kähler metric on the complement of a divisor constructed by Tian-Yau
should be the limit of the conic Kähler.Einstein metrics as the angle 2πβ tends
to 0. More recently, in [Do10], Donaldson proposed a continuity method of using
these metrics to construct smooth Kähler-Einstein metrics on Fano manifolds
by deforming the cone angle of conic Kähler-Einstein metrics of positive scalar
curvature.
In [Br11], S. Brendle obtained existence of Ricci-flat conic Kähler metrics
with cone angle 2πβ along a smooth D with restriction, i.e., β ≤ 1/2. In
[JMR11], Jeffres, Mazzeo-Rubinstein, together with a calculation due to C. Li
and Rubinstein (see Appendix to [JMR11]), established the existence of conic
Kähler-Einstein metrics with cone angle 2πβ < 2π and non-positive scalar curvature. They also did the case of positive scalar curvature, as stated in Theorem
6.7 below. Donaldson’s work [Do10] has played a very important role in constructing conic Kähler-Einstein metrics. By different methods but allowing D to
have simple normal crossings, in [Be11], Berman showed how to produce KählerEinstein metrics whose volume form is asymptotic to that of a conic metric by
a variational approach, while in [CGP11], Campana, Guenancia and Paun gave
an existence on conic Kähler-Einstein metrics with non-positive scalar curvature
and cone angle 2πβ < π. However, those approaches do not give desired information about the regularity of the metric along the divisor as Jeffres-MazzeoRubinstein have (also [Br11] for weaker results). It remains to consider the case
of conic Kähler-Einstein metrics with positive scalar curvature.11
11 Of
course, one can also study conic Kähler metrics of constant scalar curvature, or more
20
In rest of this section, we consider the positive case. Let M be a Fano
manifold with a smooth divisor D ∈ | − KM |. We want to look for KählerEinstein with conic singularities along D with angle 2πβ. This is equivalent to
solving the following equation.
Ric(ωβ ) = βωβ + (1 − β){D} ⇐⇒ (ω +
√
¯ n = ehω −βϕ
−1∂ ∂ψ)
ωn
|s|2(1−β)
As in the smooth case, there are obstructions to the existence. First one can
define the log-Futaki invariant following [Do10].
Definition 6.1. If V is a holomorphic vector field on M tangent to D. Let ω
be a smooth Kähler form in c1 (L), and θV = LV − ∇V . Define
F (c1 (L), (1 − β)D)(V )
=
F (c1 (L))(V ) + (1 − β)
(∫
ω n−1
V ol(D)
θV
−
(n − 1)! V ol(M )
D
∫
ωn
θV
n!
M
)
(6.1)
As shown in [Li11], log-Futaki invariant is an obstruction to the existence
of conic Kähler-Einstein metrics. Since (M, D) is a log Calabi-Yau, it does not
admit any nontrivial C∗ action. To obtain the obstruction for the existence,
C. Li formulated in [Li11] an analogue of the K-stability for pairs: The log-Kstability.
Any test configuration (M, L) of (M, L) can be equivariantly embedded into
CP N × C where the C∗ action on CP N is given by an one-parameter subgroup
of SL(N + 1, C). If Y is any subvariety of M (Y = (1 − β)D in our setting),
the test configuration of (M, L) also induces a test configuration (Y, L|Y ) of
(Y, L|Y ).
Let dk , d˜k be the dimensions of H 0 (M, Lk ), H 0 (Y, Lk |Y ), and wk , w̃k be the
k
weights of C∗ action on H 0 (M0 , L|M
), H 0 (Y0 , Lk |Y0 ), respectively. Here M0
0
and Y0 are central fibers of M and Y. Then we have expansions:
dk = a0 k n + a1 k n−1 + O(k n−2 ),
d˜k = ã0 k n−1 + O(k n−2 ),
wk = b0 k n+1 + b1 k n + O(k n−1 )
w̃k = b̃0 k n + O(k n−1 )
The algebraic log-Futaki invariant of the given test configuration (M, L) is
defined by
F (M, Y, L)
=
=
2(a0 b1 − a1 b0 )
ã0
+ (−b̃0 + b0 )
a0
a0
a0 (2b1 − b̃0 ) − b0 (2a1 − ã0 )
a0
(6.2)
Definition 6.2 ([Li11]). Along a test configuration (M, L), (M, Y, L) is log-Ksemistable if F (M, Y, L) ≤ 0, further, (M, Y, L) is log-K-polystable if F (M, Y, L)
generally, extremal Kähler metrics. One can easily extend many of what we know about
smooth extremal Kähler metrics to the case of conic metrics.
21
is either negative or vanishes and the normalization of (M, Y, L) is a product
test configuration.
We say (M, Y, L) log-K-semistable (resp. log-K-polystable) if, for any integer
r > 0, (M, Y, Lr ) is log-K-semistable (log-K-polystable) along any test configuration of (M, Y, Lr ).
Remark 6.3. When Y is empty, then the log-K-stability coincides the Kstability.
∑r
−1
Example 6.4. M = CP 1 , L = KCP
1 = OCP 1 (2), Y =
i=1 αi pi . For any
i ∈ {1, · · · , r}, we choose the coordinate z on CP 1 , such that z(pi ) = ∞. Then
the holomorphic vector field V = z∂z generates the one-parameter subgroup λ(t):
λ(t) · z = t · z. As t → 0, λ(t),
∑ we get a test configuration of (M, Y ) which is
of the form (CP 1 , αi {∞} + j̸=i αj {0}). A direct computation shows that the
corresponding log-Futaki invariant is given by λ:
F (CP 1 ,
r
∑
αi pi , OCP 1 (2))(λ) = αi −
i=1
It follows that (CP 1 ,
∑r
i=1
∑
αj
j̸=i
αi pi ) is log-K-stable if and only if for any i,
∑
αj < 0.
αi −
(6.3)
j̸=i
The log-K-stability is related to the problem of constructing conic metric of constant scalar curvature on CP 1 with conic angle 2π(1−αi ) at pi . As in the case of
Kähler-Einstein metrics, the only non-trivial case is that of positive curvature.
It follows from the works of Troyanov, McOwen, Thurston and Luo-Tian
Theorem 6.5 (See [LT92] and the references there). There is a unique conic
metric on CP 1 with prescribed conic angles at pi as above and of constant curvature if and only if
∑r
(a)
i=1 αi < 2, and
∑
(b) αi − j̸=i αj < 0, for all i = 1, . . . , n.
Condition (a) assures that the curvature is positive, while Condition (b)
means exactly the lon=K-stability.
This example is an evidence of the following conjecture:
Conjecture 6.6 (Logarithmic version of Tian-Yau-Donaldson conjecture). There
is a conic Kähler metric of constant scalar curvature on (M, Y ) if and only if
(M, Y ) is log-K-stable.
One can also integrate the log-Futaki-invariant to get the log-K-energy:
(
(∫
) ∫
) n
∫
ωϕn
ωϕn
ωϕ
ωn
0
Tω,(1−β)D =
log
+β
ϕ
+
F
(ϕ)
+
h
ω
ω
n!
n!
ehω ω n /|s|2(1−β) n!
M
M
M
The following is an extension of the existence theorem on smooth KählerEinstein metrics to the conic case.
22
Theorem 6.7 ([JMR11]). If the log-K-energy is proper, then there exists a
conic Kähler-Einstein metric.
One can also prove
Theorem 6.8. ([LS12]) If the log-K-energy Tω,(1−β)D is bounded from below
(resp. proper), the polarized pair ((M, (1 − β)D), −KM ) is log-K-semistable
(resp. log-K-stable).
Corollary 6.9. If there exists a conic Kähler-Einstein metric on (M, (1−β)D),
then ((M, (1 − β)D), −KM ) is log-K-semistable.
References
[ALV09] Arezzo, C., Lanave, G. and Vedova. A.: Singularities and Ksemistability. arXiv:0906.2475.
[ArPa05] Arezzo, C. and Pacard, F.: Blowing up and desingularizing constant
scalar curvature Kähler manifolds. Acta Math. 196 (2006), 179-228.
[Au76] Aubin, T.: Equations du type de Monge-Ampère sur les variétés
Kähleriennes compactes. C. R. Acad. Sci. Paris, 283 (1976), 119 - 121.
[Au83] Aubin, T.: Réduction du cas positif de l’equation de Monge-Ampère sur
les varietes Kahleriennes compactes á la demonstration d’une inegalite.
J. Funct. Anal., 57 (1984), 143–153.
[Be11] Berman, R.: A thermodynamical formalism for Monge-Amp‘ere
equations, Moser-Trudinger inequalities and Kahler-Einstein metrics.
arXiv:1011.3976.
[BC65] Bott, R. and Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114 1965
71112.
[BM86] Bando, S. and Mabuchi, T.: Uniqueness of Einstein Kähler metrics modulo connected group actions. Algebraic Geometry, Adv. Studies in Pure
Math., 10 (1987).
[Br11] Brendle, S.: Ricci flat Kahler metrics with edge singularities.
arXiv:1103.5454.
[CGP11] Campana, F., Guenancia, H. and Paun, M.: Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields.
arXiv:1104.4879.
[Cal82] Calabi, E.: Extremal Kähler metrics. Seminar on Diff. Geom., Ann. of
math. Stud., 102, Princeton Univ. Press, 1982.
23
[Cal85] Calabi, E.: Extremal Khler metrics, II. Differential geometry and complex analysis, 95114, Springer, Berlin, 1985.
[Cat]
Catlin, D.: The Bergman kernel and a theorem of Tian. Analysis and geometry in several complex variables (Katata, 1997), 123, Trends Math.,
Birkhuser Boston, 1999.
[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.
[Ch00] Chen, X.X.: The space of Khler metrics. J. Differential Geom. 56 (2000),
no. 2, 189234.
[ChLiSh] Chen, Bohui, Li, Anmin and Shen Li: Uniform K-stability for extremal
metrics on toric varieties. arXiv:1109.5228.
[ChSo10] Chen, X.X. and Sun, S.: Calabi flow, Geodesic rays, and uniqueness
of constant scalar curvature Kähler metrics. math.DG/1004.2012.
[ChTi] Chen, X.X and Tian, G.: Geometry of Kähler metrics and foliations
by holomorphic discs. Publ. Math. Inst. Hautes tudes Sci. 107 (2008),
1–107.
[DiTi92] Ding, W. and Tian, G.: Kähler-Einstein metrics and the generalized
Futaki invariants. Invent. Math., 110 (1992), 315-335.
[Do00] Donaldson, S: Scalar curvature and projective embeddings. I. J. Diff.
Geom. 59 (2001), 479-522.
[Do02] Donaldson, S: Scalar curvature and stability of toric varieties. J. Diff.
Geom., 62 (2002), 289-349.
[Do05] Donaldson, S: Lower bounds on the Calabi functional, J. Diff. Geom.,
70 (2005), 453-472.
[Do10] Donaldson, S: Stability, birational transformations and the KahlerEinstein problem. arXiv:1007.4220.
[Fut83] Futaki, A: An obstruction to the existence of Einstein-Kähler metrics.
Inv. Math., 73 (1983), 437-443.
[Fut90] Futaki, A: Kähler-Einstein Metrics and Integral Invariants. Lecture
Notes in Mathematics, 1314, Springer-Verlag.
[JMR11] Jeffres, T., Mazzeo, R. and Rubinstein, Y.: Khler-Einstein metrics
with edge singularities. arXiv:1105.5216.
[Li09] Li, Chi: Constant scalar curvature Khler metric obtains the minimum of
K-energy. Int. Math. Res. Not. IMRN 2011, no. 9, 21612175.
24
[Li11] Li, Chi: Remarks on logarithmic K-stability. arXiv:1104.0428.
[LS12] Li, Chi and Sun, S.: In preparation.
[LX11] Li, Chi and Xu, CY: Special test configurations and K-stability of Fano
varieties. arXiv:1111.5398.
[Lu00] Lu, Zhiqin: On the lower order terms of the asymptotic expansion of
Tian-Yau-Zelditch. Amer. J. Math. 122 (2000), no. 2, 235273.
[LT92] Luo, F. and Tian, G.: Liouville equation and spherical convex polytopes.
Proc. Amer. Math. Soc. 116 (1992), no. 4, 11191129.
[Ma86] Mabuchi, T: Some symplectic geometry on compact Kähler manifolds.
Osaka J. Math., bf 24 (1987), 227-252.
[Paul08] Paul, S.: Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi
energy asymptotics. Ann. of Math. (2) 175 (2012), no. 1, 255296.
[PT04] Paul, S. and Tian, G.: CM Stability and the Generalized Futaki Invariant I. math.DG/ 0605.278.
[PT06] Paul, S. and Tian, G.: CM stability and the generalized Futaki invariant
II. Astrisque No. 328 (2009), 339354.
[Ru07] Rubinstein, Y.: On energy functionals, Kähler-Einstein metrics, and the
Moser-Trudinger-Onofri neighborhood, J. Funct. Anal. 255 (2008), 26412660.
[Ru97] Ruan, Wei-dong: Canonical coordinates and Bergmann metrics. Comm.
Anal. Geom. 6 (1998), no. 3, 589631.
[Sto07] Stoppa, J: K-stability of constant scalar curvature Kähler manifolds.
Adv. Math. 221 (2009), 1397-1408.
[Ti87] Tian, G.: On Kähler-Einstein metrics on certain Kähler Manifolds with
C1 (M ) > 0. Invent. Math., 89 (1987), 225-246.
[Ti88] Tian, G.: Harvard PhD Thesis, May, 1988.
[Ti89] Tian, G.: On Calabi’s conjecture for complex surfaces with positive first
Chern class. Inv. Math. 101, (1990), 101-172.
[Ti90] Tian, G: Kähler-Einstein on algebraic manifolds. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 587598,
Math. Soc. Japan, Tokyo, 1991.
[Ti94] Tian, G: Khler-Einstein metrics on algebraic manifolds. Transcendental
methods in algebraic geometry (Cetraro, 1994), 143185, Lecture Notes
in Math., 1646, Springer, Berlin, 1996.
25
[Ti97] Tian, G: Kähler-Einstein metrics with positive scalar curvature. Invent.
Math., 130 (1997), 1-39.
[Ti98] Tian, G: Canonical Metrics on Kähler Manifolds. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, 2000.
[Ti09] Tian, G: Einstein metrics on Fano manifolds. To appear in the proceeding for the 2008 conference celebrating J. Cheeger’s 65th birthday.
[Ti12] Tian, G: Partial C 0 -estimates. In preparation.
[TY87] Tian, G. and Yau, S.T.: Kähler-Einstein metrics on complex surfaces
with C1 (M ) positive. Comm. Math. Phys., 112 (1987).
[TZ97] Tian, G and Zhu, X.H.: A nonlinear inequality of Moser-Trudinger type.
Calc. Var. Partial Differential Equations, 10 (2000), 349-354.
[TZ00] Tian, G. and Zhu, X.H.: Uniqueness of Kähler-Ricci solitons. Acta Math.
184 (2000), no. 2, 271-305.
[TZ07] Tian, G. and Zhu, X.H.: Convergence of Kähler-Ricci flow. J. Amer.
Math. Soc., 20 (2007), 675-699.
[Ya78] Yau, S.T: On the Ricci curvature of a compact Kähler manifold and
the complex Monge-Ampére equation, I. Comm. Pure Appl. Math., 31
(1978), 339-441.
[Ze]
Zelditch, S.: Szego kernels and a theorem of Tian. Internat. Math. Res.
Notices 1998, no. 6, 317331.
[ZZ08] Zhou, Bin and Zhu, Xiaohua: Relative K -stability and modified K energy on toric manifolds. Adv. Math. 219 (2008), no. 4, 13271362.
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