A Bando-Mabuchi Uniqueness Theorem

0.2 MB·English

by  Li Yi

A Bando-Mabuchi Uniqueness Theorem
Li YI
Institut Elie Cartan, Nancy
3
Abstract. In this paper we prove a uniqueness theorem on generalized
1
0 K¨ahler-Einstein metrics on Fano manifolds. Our result generalize the one
2
shown by Berndtsson using the convexity properties of Bergman kernels. The

n
same technics as well as that of the regularization of closed positive currents
a
J and an estimate due to Demailly-Koll´ar will be used in our proof.

5
1
§0. Introduction.

]
V
C Let X be an n-dimensional projective manifold, here n 3; we denote by
≥
. c (X) its first Chern class, i.e. the Chern class corresponding to the anti
h 1
t canonical bundle K . Let α be a (1,1) class on X; here we denote by
a − X { }
m α a non-singular representative. We assume that the following requirements
[ are satisfied.

2
(i) The class c (X) α is Ka¨hler, i.e. it contains a Ka¨hler metric which
v 1
−{ }
7 we denote by ω;
4
8
2 (ii) The class α is pseudo-effective; let Θ α be a closed positive
{ } ∈ { }
. current;
1
0
3
(iii) If we denote Ric(ω) the Ricci curvature associated to the metric ω,
1
: then we have
v
Ric(ω) = ω +Θ √ 1∂∂f
i Θ
X − −
for some function f which is unique up to an additive constant. We
r Θ
a
assume that there exists a real number ε > 0 such that
0
e−2(1+ε0)fΘdV < .
Z ∞
X
In other words, the singularities of the current Θ are not too wide.
Following Bedford and Taylor (c.f. [5]), if φ is a psh bounded function defined
on an open subset Ω Cn, then the quantity
⊂
√ 1∂∂φ n
−
(cid:0) (cid:1)
1
is a positive measure of locally finite mass on Ω. Hence it makes sense to
consider the equation
(⋆) (ω +√ 1∂∂ϕ)n = e−ϕ−fΘωn
−
where the solution ϕ is assumed to be bounded, and such that
ω := ω +√ 1∂∂ϕ
ϕ
−
is a Ka¨hler current (i.e. it is greater than a Ka¨hler metric).
We remark that if Θ is non-singular, then the equation (⋆) is equivalent
with the following identity
Ric(ω ) = ω +Θ.
ϕ ϕ
The uniqueness of the solution ϕ up to a biholomorphism of X was estab-
lished by Bando-Mabuchi in [4]. If Θ is the current of integration on a divisor
D, such that the pair (X,D) is klt, then the question was settled by Berndts-
son in [2], who found a proof within the framework of the psh variation of
Bergman kernels.
In this article we show that the following generalization of the result in [2]
holds.
Theorem 1. Let ϕ and ϕ be two bounded solutions of the equation (⋆).
0 1
Then there exists a biholomorphism F : X X such that
→
F (ω ) = ω and F (Θ) = Θ.
∗ ϕ1 ϕ0 ∗
If we assume that the anti-canonical class has further positivity proper-
ties, for example, K 0, then one proves in [2] that the proceeding result
X
− ≥
holds under the assumption e−fΘ ∈ L1loc(X). However, we remark that in
order to have this result, we have to assume two strong positivity properties:
c (X) α is a Ka¨hler class, and c (X) is a semi-positive class.
1 1
−{ }
The arguments we will invoke in order to prove this result generalize the
approach by Berndtsson in [2]. It is based on the convexity properties of
Bergman kernel, which we recall next.
2
Let L X be a line bundle, which is endowed with a family of non-
→
singular metrics h = e ϕt, t D, the unit disk. We assume that the function
t −
∈
ϕ(t,x) = ϕ (x)
t
is non-singular and psh. We can see L as a bundle over the family X D;
×
the bundle structure is independent of t D, but the hermitian structure
∈
varies with respect to t. Let
E := p (K +L)
X D/D
∗ ×
be the direct image of the relative adjoint bundle of L, and let (ut)t D be a
∈
holomorphic section of E. Of course, in general (u ) does not correspond to
t
a holomorphic n-form on X D : we can only represent (ut)t D by an n-form
U on X D such that ∂U×is equal to some multiple of dt∈, and such that
×
U = u . In order to evaluate the curvature Θ of E in the direction of
X t t E
| ×{}
the section (u ), we introduce a family (v ) of (n 1)-forms with values in L
t t
−
such that
∂v ω = 0,
t
∗ ∧
∂ϕtv = P (ϕ˙ u )
t t t
∗ ⊥
where P is the projection on the orthogonal complement of the holomorphic
n-forms on each fiber, then we have the following formula
Θ u ,u = p (c √ 1∂∂ϕ u ue ϕt)+ ∂v 2e ϕt√ 1dt dt
E t t n t − t −
h i ∗ − ∧ ∧ Z k k − ∧
X
b b
where we denote by u = (u dt v ) , another representation of our initial
t t t
− ∧
section (u ).
t
b
We assume now that for some geometric reasons, the curvature of E
vanishes. As a consequence, v will be holomorphic, as well as the first term
t
on the right hand side of the formula above. But then we can define a vector
field by the formula
v = V u ;
t t t
− ⌋
it will be holomorphic outside u = 0. Actually, under the condition Θ = 0
t E
one can see that V is the complex gradient of √ 1∂ϕ˙ ; finally, a simple com-
t t
−
putation shows that the flow associated to V , F, is the holomorphic map we
1
3
seek.
While trying to implement this scheme in our setting, we have to face two
important difficulties. In our case, the bundle L will be just K , endowed
X
−
with the following family of metrics. We connect ϕ and ϕ with a geodesic,
0 1
(this is known to exist), and this will give us a metric ϕ whose curvature
t
ω = ω +√ 1∂∂ ϕ
t X t
−
belongs to the class c (X) α . Then the current
1
−{ }
ω +Θ
t
with Θ α is closed, positive, and belongs to c (X) for each t. Our main
1
∈ { }
concerns are:
Θ is not necessarily smooth;
∗
the current ω = ω+√ 1∂∂ ϕ is not necessarily Ka¨hler, it is only known
t X t
∗ −
to be positive.
The plan of our article will be as follows: we approximate ω and Θ with
t
non-singular objects (section 1), and then we construct the analogue vν of
t
the (n 1)-form v mentioned above. Thanks to the fact that ϕ and ϕ
t 0 1
−
are solutions of (⋆), we show that the curvature of E, endowed with the
regularized metric, converges to zero. An uniformity argument (which is
quite subtle, given that the Lelong set of Θ could be quite complicated)
shows that vν converges to a holomorphic (n 1)-form. Then the vector field
t −
will be defined as above.
§1. Regularization process.
Fix a non-singular real (1,1)-form ω in the Ka¨hler class c (X) α , a
1
− { }
quasi-psh function ϕ is said to be ω-psh if ω + √ 1∂∂ϕ 0. We denote
− ≥
PSH(X,ω) L the convex space of all ω-psh functions which are bounded
∞
∩
on X. Let ϕ and ϕ PSH(X,ω) L , we call the path of semi-positive
0 1 ∞
∈ ∩
forms
ω = ω +√ 1∂∂ϕ witht [0,1]
t t
− ∈
a (generalized) geodesic in PSH(X,ω). The following theorem implies the
existence of a continuous geodesic which connects ω and ω (cf. [3]):
0 1
4
Theorem 2. Assume that the semi-positive closed (1,1)-forms ω and ω
0 1
belong to the same K¨ahler class c (X) α and have bounded coefficients.
1
−{ }
Then there exists a geodesic ω connecting ω and ω with the properties
t 0 1
that it is continuous on [0,1] X and that there is a constant C such that
×
ω Cω, i.e. ω has uniformly bounded coefficients.
t t
≤
Remark 3. The geodesic ω can be constructed in such a way that it is
t
Lipschitz in t. Indeed, like shown in [2], define first
( ) ϕ = sup κ
t t
∗ { }
where the supremum is taken over all psh κ with
t
lim κ ϕ ;
t 0,1
t 0,1 ≤
→
then construct a barrier
χ = max(ϕ ARet,ϕ +A(Ret 1))
t 0 1
− −
with A sufficiently large. This χ is psh and we have
t
lim χ = ϕ .
t 0,1
t 0,1
→
Therefore the supremum in ( ) has the same property if we restrict to those
∗
κ which are larger that χ. For such κ the onesided derivative at 0 is larger
than A and the onesided derivative at 1 is smaller than A. We assume from
−
now on that κ is independent of the imaginary part of t, κ is convex in t so
the derivative with respect to t increases, and must lie between A and A.
−
Therefore ϕ satisfies
t
ϕ ARet ϕ ϕ +ARet
0 t 0
− ≤ ≤
at 0 and a similar estimate at 1. Thus
lim ϕ = ϕ
t 0,1
t 0,1
→
uniformly on X. In addition, the upper semicontinuous regularization ϕ of
∗t
ϕ must satisfy the same estimate. Since ϕ is psh it belongs to the class of
t ∗t
competitors for ϕ and must coincide with ϕ , so ϕ is psh.
t t t
5
Our proof begins with the regularization for ϕ , the ampleness of c (X) α
t 1
−{ }
allows us to write ϕ as a uniform limit of a decreasing sequence of non-
t
singular functions ϕν with the property that
t
ων := ω +√ 1∂∂ϕν 0.
t − t ≥
In order to regularize the current
Θ = α+√ 1∂∂θ,
−
we use the following result:
Theorem 4. ([7]) Let T be a closed almost positive (1,1)-current and let α
be a non-singular real (1,1)-form in the same ∂∂-cohomology class as Θ, i.e.
Θ = α+√ 1∂∂θ where θ is an almost psh function. Let γ be a continuous
−
real (1,1)-form such that Θ γ. Suppose that (1) is equipped with a
≥ OTX
non-singular hermitian metric such that the Chern curvature form satisfies
c( (1))+π u 0
OTX X∗ ≥
with π : P(T X) X and with some nonnegative non-singular (1,1)-form
X ∗
→
u on X. Fix a hermitian metric ω0 on X. Then for every c > 0, there is a
sequence of closed almost positive (1,1)-current Θ = α+√ 1∂∂θ such
c,k c,k
−
that θ is non-singular on X E (Θ) and decreases to θ as k tends to +
c,k c
\ ∞
(in particular, the current Θ is non-singular on X E (Θ) and converges
c,k c
\
weakly to Θ on X) and such that
(1) Θ γ min λ ,c u ε ω0 where:
c,k k k
≥ − { } −
(2) λ (x) is a decreasing sequence of continuous functions on X such that
k
lim λ (x) = ν(Θ,X) at every point.
k
k +
→ ∞
(3) ε is positive decreasing and lim ε = 0.
k k
k +
→ ∞
(4) ν(Θ ,x) = (ν(Θ,x) c) at every point x X.
c,k +
− ∈
Recall that
E (Θ) = x X;ν(Θ,x) c
c
{ ∈ ≥ }
6
is the c-upperlevel set of Lelong numbers of Θ. A well-known theorem of Siu
[15] asserts that E (Θ) is an analytic subset of X for any c > 0.
c
Choose u = Aω0 where A is a positive constant large enough and choose
c = 1 with ν N a decreasing sequence, according to the above theorem, we
ν ∈
get a sequence of functions θν such that they are non-singular on X E (Θ)
1
\ ν
and
1
Θν := α+√ 1∂∂θν Amin λ , ω0 ε ω0
ν ν
− ≥ − { ν} −
holds on X for every ν, where λ (x) is a decreasing sequence of continuous
ν
functions on X such that lim λ (x) = ν(Θ,x) at every point and ε are
ν ν
ν +
positive decreasing such tha→t ∞lim ε = 0.
ν
ν +
→ ∞
During the proof we need to use the ”metric” form ”e ” concerning the
−·
currents ω and Θ. So we revise next the notations. Denote φ = φ0+ϕ with
t t t
√ 1∂∂φ0 = ω. So ω can be written as
t
−
ω = √ 1∂∂φ .
t t
−
Similarly, denote ψ = ψ0 +θ with √ 1∂∂ψ0 = α. Then Θ can be presented
−
as
Θ = √ 1∂∂ψ.
−
Hence if we denote τ = φ + ψ and respectively τν = φν + ψν, the above
t t t t
discussions imply that the sequence of currents
ων +Θν = √ 1∂∂τν
t − t
decreases to
ω +Θ = √ 1∂∂τ
t t
−
as ν tends to + and is non-singular on X E (Θ). We evaluate next the
1
lower bounds of∞ων + Θν. To this end, deno\teνfirst a neighborhood set of
t
E (Θ) by
1
ν
U = x X : eψν < δ
δν { ∈ ν}
(the choice of δ to beprecise ina moment). Remark that according to Theo-
ν
rem 2, ω has uniformly bounded coefficients, since ων converge uniformly to
t t
7
ωt, we have actually eτtν < Cδν for x ∈ Uδν with C > 0 a numerical constant.
Then by Theorem 4, we have
1
√ 1∂∂τν Amin λ , +ε ω0 (1)
− t ≥ −(cid:18) { ν ν} ν(cid:19)
1
on X U . As ν + , the quantity Amin λ , +ε ω0 tends to
\ δν → ∞ −(cid:18) { ν ν} ν(cid:19)
zero; Globally on X, we have
1
√ 1∂∂τν 2A ω0 3Aω0. (2)
− t ≥ −(cid:18) ν(cid:19) ≥ −
In the following proof we need to solve some ∂τtν-equations, to insure that the
solutions of such equations we will deal with satisfy certain L2-properties, we
need to modify further our approximations in order that the new ”metrics”
are non-singular on X. To this end, define
1
τν = log eτtν + .
t (cid:18) ν(cid:19)
e
1
Since eτtν + > 0, functions τν constructed in this way are therefore non-
ν t
singular on X. We analyze next their Hessian properties.
e
1
√ 1∂∂log eτtν +
− (cid:18) ν(cid:19)
eτtν∂τν
=√ 1∂ t
− eτtν + 1
ν
(eτtν∂∂τν +eτtν∂τν ∂τν)(eτtν + 1) e2τtν∂τν ∂τν
=√ 1 t t ∧ t ν − t ∧ t
− (eτtν + 1)2
ν
(eτtν + 1)eτtν∂∂τν + 1eτtν∂τν ∂τν
=√ 1 ν t ν t ∧ t
− (eτtν + 1)2
ν
eτtν 1eτtν
=√ 1 ∂∂τν +√ 1 ν ∂τν ∂τν.
− eτtν + 1 · t − (eτtν + 1)2 · t ∧ t
ν ν
We hope that Hessians of these new functions τν have similar properties as
t
τν, that is,
t
√ 1∂∂τν e
− t
e
8
would be uniformly bounded from below on X and almost positive (in other
words, thelower boundstendtozeroasν tendstoinfinity) onX U .Indeed,
\ δν
this will be true if we choose properly the value of δ . Observe first that we
ν
have
eτtν
0 < 1,
eτtν + 1 ≤
ν
so the first term in the last equality satisfies our requirements. We now
analyze the second term. Notice that we have
√ 1∂τν ∂τν = ∂τν 2 0
− t ∧ t | t | ≥
and
1eτtν
ν 0 (3)
(eτtν + 1)2 ≥
ν
hold on X. So it is sufficient to check that the function in (3) goes to zero on
X U . Indeed, for x X U , we have
\ δν ∈ \ δν
1eτtν 1 eτtν 1 1 C
ν = .
(eτtν + ν1)2 ≤ ν e2τtν ν eτtν ≤ νδν
1
Hence if we choose for example δ = , we obtain that
ν
√ν
C C
=
νδ √ν
ν
tends to zero as ν tends to infinity. Therefore the second term fulfills also
our requirements. We can conclude from (1), (2) and the above discussions
that
√ 1∂∂τν A ω0 (4)
− t ≥ − 1
on X with A > 0 a numerical constant as well as
1 e
√ 1∂∂τν A ω0 (5)
− t ≥ − δν,ν
on X U with the property that eA tends to zero as ν + . For the
\ δν δν,ν → ∞
sake of simplifying the notations, we denote from now on τν still by τν with
t t
the understanding that it is now a sequence of non-singular functions on X
e
satisfying (4) and (5) on the relevant sets.
9
§2. Constructions of vν.
t
Like explained in the introduction, finding solutions of the ∂τtν-equations
∂τtνvν = Pν,t(τ˙νu ) (6)
t t t
⊥
is important in the search of holomorphic vector fields. In this expression, u
t
is the section of E described in the introduction,
∂τν
τ˙ν = t
t ∂t
and Pν,t is the projection on the orthogonal complement of the holomorphic
n-form⊥s on each fiber, and finally the operator ∂τtν is defined by
∂τtν := eτtν∂e−τtν = ∂ −∂τtν ∧.
We will explain why equation (6) can always be solved for τν fixed in a
t
moment. In order to analyze the uniformity properties of these solutions,
the following lemma will be important (it is a slight variation of Lemma 6.2
in [2]).
Lemma 5. Let L be a holomorphic line bundle over X with a metric ξ
satisfying √ 1∂∂ξ c ω with c > 0 a numerical constant. Let ξ be a
0 0 0
− ≥ −
smooth metric on L with ξ ξ and assume
0
≤
I := eξ0 ξ < .
−
Z ∞
X
Then there is a constant B, only depending on I and ξ such that if f is a
0
∂-exact L valued (n,1)-form with
f 2e ξ 1, (7)
−
Z | | ≤
X
there is a solution u to ∂u = f with
u 2e ξ B.
−
Z | | ≤
X
10