On the Kahler Ricci flow on projective manifolds of general type

0.28 MB·English

by  Bin Guo

ON THE KA¨HLER RICCI FLOW ON PROJECTIVE MANIFOLDS OF
GENERAL TYPE
BIN GUO
5
1 Abstract. We consider the K¨ahler Ricci flow on a smoothminimal model of generaltype,
0 we show that if the Ricci curvature is uniformly bounded below along the K¨ahler-Ricci
2 flow, then the diameter is uniformly bounded. As a corollary we show that under the Ricci

n curvaturelowerboundassumption,theGromov-Hausdorfflimitoftheflowishomeomorphic
a to the canonical model. Moreover, we can give a purely analytic proof of a recent result of
J Tosatti-Zhang([29])thatifthecanonicallinebundleKX isbigandnef,butnotample,then
7
the flow is of Type IIb.
1

]
G 1. Introduction
D
The Ricci flow ([10]) has been one of the most powerful tools in geometric analysis with
.
h
remarkable applications to the study of 3-manifolds. The complex analogue, the Ka¨hler-
t
a Ricci flow, has been used by Cao ([2]) to give an alternative proof of the existence of Ka¨hler
m
Einstein metrics on manifolds with negative or vanishing first Chern class ([31, 1]). Tsuji
[
[30] applied the Ka¨hler-Ricci flow to construct a singular Ka¨hler Einstein metrics on smooth

1
minimal manifolds of general type. The analytic Minimal Model program, introduced in
v
9 [19, 20], aims to find the minimal model of an algebraic variety, by running the Ka¨hler-Ricci
3 flow. It is conjectured ([20]) that the Ka¨hler-Ricci flow will deform a given projective variety
2
to its minimal model and eventually to its canonical model coupled with a canonical metric
4
0 of Einstein type, in the sense of Gromov-Hausdorff.
.
1 Let X be a projective n-dimensional manifold, with the canonical bundle KX big and nef.
0 We consider the Ka¨hler-Ricci flow
5
1 ∂ω
: (1.1) = Ric(ω) ω, ω(0) = ω0,
v ∂t − −
i
X
where ω is a Ka¨hler metric on X. It’s well-known that the equation (1.1) is equivalent to
0
r the following complex Monge-Ampere equation
a
∂ϕ (χ+e−t(ω χ)+i∂∂¯ϕ)n
0
= log − ϕ
(1.2)  ∂t Ω −

ϕ(0) = 0,

¯
where Ω is a smooth volume form, χ = i∂∂logΩ c (K ) = c (X), and ω(t) = χ +
1 X 1
e−t(ω χ) + i∂∂¯ϕ. It’s also well-known ([30, 27])∈that the equ−ation (1.2) has long time
0
−
existence, if K is nef. We will prove the following result:
X
Theorem 1.1. Let X be a projective manifold with K big and nef. If along the K¨ahler
X
Ricci flow (1.1), the Ricci curvature is uniformly bounded below for any t 0, i.e.,
≥
Ric(ω(t)) Kω(t),
≥ −
1
2 BIN GUO
for some K > 0, then there is a constant C > 0 such that the diameter of (X,ω(t)) remain
bounded, i.e.,
diam(X,ω(t)) C.
≤
Remark 1.1. If we use Kawamata’s theorem ([11]) that the nef and big canonical line bundle
K is semi-ample, by [32, 21] the scalar curvature along the K¨ahler Ricci flow (1.1) is uni-
X
formly bounded, hence Ricci curvature lower bound implies that Ricci curvature is uniformly
bounded on both sides. Then in the proof of Theorem 1.1, we can use Cheeger-Colding-Tian
([4]) theory to identify the regular sets. Moreover, if K is semi-ample and big, then the L∞
X
bound of ϕ in (1.2) will simplify the proof. However, following Song’s ([18]) recent analytic
proof of base point freeness for nef and big K , our proof of Theorem 1.1 does not rely on
X
Kawamata’s theorem.
It is conjectured by Song-Tian in [20] that the Ka¨hler Ricci flow (1.1) will converge to the
the canonical model of X coupled with the unique Ka¨hler Einstein current with bounded
potential, in the Gromov-Hausdorff sense. Under the assumption that the Ricci curvature is
uniformly bounded below, we can partially confirm this conjecture.
Corollary 1.1. Under the same assumptions as Theorem 1.1, then as t ,
→ ∞
(X,ω(t)) dGH (X ,d ),
∞ ∞
−−→
the limit space X is homeomorphic to the canonical model X of X. Moreover, (X ,d )
∞ can ∞ ∞
is isometric to the metric completion of (X◦ ,g ), where g is the unique K¨ahler-Einstein
can KE KE
current with bounded local potentials and X◦ is the regular part of X .
can can
Consider the unnormalized Ka¨hler Ricci flow
∂
(1.3) ω = Ric(ω), ω(0) = ω ,
0
∂t −
with long time existence. The flow (1.3) is called to be of Type III, if
sup t Rm (x,t) < ,
| | ∞
X×[0,∞)
otherwise it is of Type IIb, here Rm (ω(t)) denotes the Riemann curvature of ω(t). It’s well-
| |
known that Type III condition is equivalent to the curvature is uniformly bounded along the
normalized Ka¨hler Ricci flow (1.1). As a by-product of our proof of Theorem 1.1, we obtain
a purely analytic proof of a recent result of Tosatti-Zhang ([29]), namely,
Theorem 1.2. [29] Let X be a projective manifold with K big and nef, if the K¨ahler Ricci
X
flow (1.1) is of type III, then the canonical line bundle K is ample.
X
Throughout this paper, the constants C may be different from lines to lines, but they are
all uniform. We also use g as the associated Riemannian metric of a Ka¨hler form ω, for
example the metric space (X,ω(t)) means the space (X,g(t)).
Acknowledgement: The author would like to thank his advisor Prof. J. Song, for his
constant help, support and encouragement over the years. He also wants to thank the
members of the complex geometry and PDE seminar at Columbia University, from whom he
learns a lot. His thanks also goes to Prof. V. Tosatti for his careful reading of a preprint of
this paper and many helpful suggestions which made this paper clearer. Finally he likes to
thank Prof. X. Wang and V. Datar for their interest and helpful discussions.
ON THE KA¨HLER RICCI FLOW ON PROJECTIVE MANIFOLDS OF GENERAL TYPE 3
2. Preliminaries
In this section, we will recall some definitions and theorems we will use in this note.
Definition 2.1. Let L X be a holomorphic line bundle over a projective manifold X. L is
said to be semi-ample if→the linear system kL is base point free for some k Z+. L is said
| | ∈
to be big if the Iitaka dimension of L is equal to the dimension of X. L is called numerically
effective (nef) if L C 0 for any irreducible curve C X.
· ≥ ⊂
We can define a semi-group
(X,L) = k Z+ kL is base point free .
F { ∈ | }
For any k (X,L), the linear system kL induces a morphism
∈ F | |
Φ = Φ : X X = Φ (X) PNk
k |kL| k k
→ ⊂
where N + 1 = dimH0(X,kL). It’s well-known that ([13]) that for large enough k,l
k
∈
(X,L), (Φ ) = , Φ = Φ and X = X .
F k ∗OX OXk k l k l
We will need the following version of L2 estimates due to Demailly ([7])
Theorem 2.1. Suppose X is an n-dimensional projective manifold equipped with a smooth
K¨ahler metric ω. Let L be a holomorphic line bundle over X equipped with a possibly singular
hermitian metric h such that Ric(h) + Ric(ω) δω in the current sense for some δ > 0.
≥
Then for any L-valued (0,1)-form τ satisfying
∂¯τ = 0, τ 2 ωn < ,
Z | |h,ω ∞
X
¯
there exists a smooth section u of L such that ∂u = τ and
1
u 2 ωn τ 2 ωn.
Z | |h,ω ≤ 2πδ Z | |h,ω
X X
3. Identify the regular sets
Since K is big and nef, by Kodaira lemma, there exists an effective divisor D X, such
X
⊂
that K εD is ample, hence there exists a hermitian metric on [D] such that
X
−
χ εRic(h ) > 0.
D
−
Let’s recall a few known estimates of the flow, (see [15, 30] or [17, 18] without assuming
that K is semi-ample)
X
Lemma 3.1. (i) There is a constant C > 0 such that for any t 0,
≥
∂
supϕ(t) C, supϕ˙(t) = sup ϕ C.
≤ ∂t ≤
X X X
(ii) For any δ (0,1), there is a constant C > 0 such that
δ
∈
ϕ δlog σ 2 C ,
≥ | D|hD − δ
where σ is a holomorphic section of the line bundle [D] associated to the divisor D.
D
(iii) Along the K¨ahler Ricci flow, there exist constants C > 0,λ > 0 such that
tr ω(t) C σ −2λ
ω0 ≤ | D|hD
4 BIN GUO
(iv) For any compact subset K X D, any ℓ Z+, there exists a constant C > 0 such
ℓ,K
⊂ \ ∈
that
ϕ C .
Cℓ(K) ℓ,K
k k ≤
Hence we can conclude that
C∞(X\D)
ω(t) loc ω ,
∞
−−−−−−→
for some smooth Ka¨hler metric ω on X D. On the other hand, it can be shown that
∞
\
ϕ˙(t) 0 on any K X D as t , hence ω satisfies the equation
∞
→ ⊂ \ → ∞
ωn = (χ+i∂∂¯ϕ )n = eϕ∞Ω, on X D,
∞ ∞ \
and ω is a Ka¨hler-Einstein metric on X D, i.e.,
∞
\
Ric(ω ) = ω .
∞ ∞
−
Proposition 3.1. [18] For any holomorphic section σ H0(X,mK ), there is a constant
X
∈
C = C(σ) such that for any t 0, we have
≥
sup σ 2 C, sup σ 2 C
| |hmt ≤ |∇t |hmt ≤
X X
where h = 1 is the hermtian metric on K induced by the K¨ahler metric ω(t) and is
t ω(t)n X ∇t
the covariant derivative with respect to hm.
t
Letting t , we have h h = h = h e−ϕ∞ on X D (here h = 1), and
→ ∞ t → ∞ KE χ \ χ Ω
(3.1) sup σ 2 C, sup σ 2 C.
| |hm∞ ≤ |∇∞ |hm∞ ≤
X\D X\D
Definition 3.1. We define a set X to be the points p X such that the µ-jets at p
X
are generated by global sections ofRmK⊂ for some m Z+, for∈any µ Nn with µ 2.
X
∈ ∈ | | ≤
Proposition 3.2. [18] is an open dense set of X and on we have locally smooth
X X
R R
convergence of ω(t) to ω .
∞
By the smooth convergence of ω(t) on X D, we can choose a point p X D and a small
\ ∈ \
r > 0 such that (we write the associated Riemannian metric of ω(t) as g(t))
0
B (p,r ) X D, Vol (B (p,r )) v , t 0
g(t) 0 g(t) g(t) 0 0
⊂⊂ \ ≥ ∀ ≥
for some v > 0. For any sequence t , (X,g(t ),p) is a sequence of almost Ka¨hler-
0 i i
→ ∞
Einstein manifolds (see the Appendix), in the sense of Tian-Wang ([24]). By the structure
theorem in Tian-Wang ([24]), we have
(3.2) (X,g(t ),p) dGH (X ,d ,p ).
i ∞ ∞ ∞
−−→
Moreover, X has a regular-singular decomposition, X = ; the singular is closed
∞ ∞
R∪S S
and of Hausdroff dimension 2n 4; the regular set is an open smooth Ka¨hler manifold,
≤ − R
and d is induced by some smooth Ka¨hler-Einstein metric g′ , i.e. on , Ric(g′ ) = g′ .
∞|R ∞ R ∞ − ∞
We define a subset X to be a set consisting of the points q X such that
X ∞ ∞
S ⊂ ∈
there exist a sequence of points q X such that q q along the Gromov-Hausdroff
k X k
∈ \R →
convergence.
By a theorem of Rong-Zhang (see Theorem 4.1 in [16]), there exists a surjective map
( ,g ) (X ,d ),
X ∞ ∞ ∞
R →
ON THE KA¨HLER RICCI FLOW ON PROJECTIVE MANIFOLDS OF GENERAL TYPE 5
where ( ,g ) denotes the metric completion of the metric space ( ,g ), and a homeo-
X ∞ X ∞
R R
morphism ( ,g ) (X ,d ) which is a local isometry.
X ∞ ∞ X ∞
It’s not hRard to see→that \Sis closed in X and any tangent cone at q is R2n, hence
X ∞ X
S 6∈ S
X , i.e., .
∞ X X
\S ⊂ R S ⊂ S
Proposition 3.3. We have
X
S ⊂ S
hence = .
X
S S
Proof. Suppose not, there exists q , then there exist q (X ,g(t )) converging
X k X k
∈ S ∩R ∈ \R
to q along the Gromov-Hausdroff convergence (3.2). Since is open and tangent cones at
points in is the Euclidean space R2n, for any small δ > 0,Rthere exists a sufficiently small
R
r > 0 such that
0
Bd∞(q,3r0) ⊂⊂ R, Volg∞′ (Bd∞(q,3r0)) > (1−δ/2)VolgE(B(0,3r0)),
where g is the standard Euclidean metric on R2n and B(0,3r ) is the Euclidean ball. Since
E 0
Ricci curvatures are bounded below, by volume continuity for the Gromov-Hausdorff conver-
gence ([6]) we have for k large enough,
Vol (B (q ,3r )) > (1 δ)Vol (B(0,3r )).
g(tk) g(tk) k 0 − gE 0
By assumption that the Ricci curvature is uniformly bounded below along the Ka¨hler Ricci
flow, hence Perelman’s pseudo-locality ([14, 24]) implies that if δ is small enough, there exists
a small but uniform constant ε > 0 such that
0
2
sup Rm(g(t +ε )) .
k 0
| | ≤ ε
Bg(tk)(qk,2r0) 0
Moreover, by Theorem 4.2 in [24],
(3.3) (B (q ,2r ),g(t +ε ),q ) dGH (B (q,2r ),d ,q).
g(tk) k 0 k 0 k −−→ d∞ 0 ∞
By Shi’s derivative estimate, we have
sup lRm(t +ε ) C(ε ,l),
k 0 0
|∇ | ≤
Bg(tk)(qk,3r0/2)
for any l N and some constant C(ε ,l). Thus we have smooth convergence of g(t + ε )
0 k 0
∈
to a Ka¨hler metric g˜ on (B (q,r ),J ) along the Gromov-Hausdorff convergence (3.3),
∞ d∞ 0 ∞
where J is the limit complex structure.
∞
Without loss of generality we can assume the injectivity radii of g(t + ε ) at q are
k 0 k
bounded below by r ([5]), since the Riemann curvatures and volumes of B (q ,r ) are
0 g(tk) k 0
uniformlybounded. Fork largeenough, thereexists(see[25])alocalholomorphiccoordinates
system z(k) n on the ball (B (q ,r ),g(t +ε )) such that z(k) 2 = n z(k) 2 r2,
{ α }α=1 g(tk) k 0 k 0 | | α=1| α | ≤ 0
z(k) 2(q ) = 0 and under these coordinates g = g ( z(k), ¯z(k)) satisPfies
| | k αβ¯ tk+ε0 ∇ α ∇ β
1
δ g Cδ , g C, for some γ (0,1).
C αβ ≤ αβ¯ ≤ αβ k αβ¯kC1,γ ≤ ∈
This implies that the Euclidean metric under these coordinates
n
(3.4) √ 1dz(k) dz¯(k)
− α ∧ α
X
α=1
6 BIN GUO
is uniformly equivalent to g(t +ε ) on the ball B (q ,r ).
k 0 g(tk) k 0
Recall that along Ka¨hler-Ricci flow
¯
Ric(ω(t)) = χ i∂∂(ϕ+ϕ˙).
− −
Take a cut-off function η on R such that η(x) = 1 for x ( ,1/2) and vanishes for
∈ −∞
x [1, ). Choose a function
∈ ∞
z(k) 2
Φ = ( µ +1+n)η | | log z(k) 2 +ϕ(t +ε )+ϕ˙(t +ε ).
k | | (cid:16) r2/2 (cid:17) | | k 0 k 0
0
Note that Φ is a globally defined function on X (with a log-pole at q ) when k is large
k k
enough.
Since the metrics (3.4) and g(t +ε ) are uniformly equivalent for k large enough on the
k 0
support of i∂∂¯ (n+1+ µ )η |z(k)|2 log z(k) 2 , we see that there is a uniform constant Λ
(cid:16) | | (cid:16) r02/2 (cid:17) | | (cid:17)
independent of k such that
z(k) 2
i∂∂¯ (n+1+ µ )η | | log z(k) 2 Λω(t +ε ).
(cid:16) | | (cid:16) r2/2 (cid:17) | | (cid:17) ≥ − k 0
0
We will fix an integer m 10Λ.
≥
Define a (singular) hermitian metric on K by
X
hk = hχe−ϕ(tk2+ε0)−mǫ log|σD|2hD,
for some small ǫ > 0. Then we have for k large enough (we denote below ω = ω(t +ε ),
k k 0
and [D] the current of integration over the divisor D.)
1
Ric(hm)+Ric(ω )+i∂∂¯Φ =mχ+ i∂∂¯ϕ ǫRic(h )+ǫ[D] χ
k k k 2 − D −
z(k) 2
+i∂∂¯ (n+1+ µ )η | | log z(k) 2
(cid:16) | | (cid:16) r2/2 (cid:17) | | (cid:17)
0
m mχ m
= ω + ǫRic(h ) e−tk−ε0(ω χ)
k D 0
2 2 − − 2 −
z(k) 2
+ǫ[D]+i∂∂¯ (n+1+ µ )η | | log z(k) 2
(cid:16) | | (cid:16) r2/2 (cid:17) | | (cid:17)
0
m
ω ,
k
≥ 4
in the current sense, for k large enough. The above inequality follows since mχ ǫRic(h )
2 − D
is a fixed Ka¨hler metric, which is greater than me−tk−ε0(ω χ) for k large enough.
2 0 −
Define an mK -valued (0,1) form
X
z(k) 2
η = ∂¯ η | | (z(k))µ ,
k,µ (cid:16) (cid:16) r2/2 (cid:17) (cid:17)
0
where
n
(z(k))µ = (z(k))µα, µ = (µ ,...,µ ) Nn.
α 1 n ∈
Y
α=1
It’s not hard to see (noting that the pole order along D is ǫ)
≤
η 2 e−Φkωn < .
Z | k,µ|hmk k ∞
X
ON THE KA¨HLER RICCI FLOW ON PROJECTIVE MANIFOLDS OF GENERAL TYPE 7
Then we can apply the Hormander’s L2 estimate (see Theorem 2.1 with L = mK ) to solve
X
¯
the following ∂-equation
¯
∂u = η ,
k,µ k,µ
with u a smooth section of mK satisfying
k,µ X
4
Z |uk,µ|hmk e−Φkωkn ≤ m Z |ηk,µ|2hmk e−Φkωkn < ∞.
X X
By checking the pole order of e−Φk at q we can see that u vanishes at q up to order µ ,
k k,µ k
| |
and hence
z(k) 2
σ := u η | | (z(k))µ
k,µ k,µ− (cid:16) r2/2 (cid:17)
0
is a nontrivial global holomorphic section of mK . Hence we see the global sections of mK
X X
generatestheµ-jetsatq fork largeenough. Thisgivesthecontradiction. Hence . (cid:3)
k X
S ⊂ S
Thus we have a local isometry homeomorphism
( ,g ) (X ,d ) = ( ,d ).
X ∞ ∞ X ∞ ∞
R → \S R
Hence we can identify and , and d is induced by the Ka¨hler-Einstein metric g .
RX R ∞|R ∞|RX
4. Estimates near the singular set
Throughout this section, we fix an effective divisor D X such that
⊂
K ε[D] > 0
X
−
for sufficiently small ε > 0. By the previous section, we see X D . Choose a log-
X
\ ⊂ R
resolution of (X,D),
π : Z X
1
→
such that π−1(D) is a smooth divisor with simple normal crossings. Fix a point O in a
1
smooth component of π−1(D) and blow up Z at the point O, we get a map
1
π : X˜ Z,
2
→
for some smooth projective manifold X˜. Denote π = π π : X˜ X.
1 2
◦ →
By Adjunction formula, we have
K = π∗K +(n 1)E +F, F = a F ,
X˜ X − k k
X
k
whereE istheexceptional locusoftheblowupπ , andF isaprimedivisor intheexceptional
2 k
locus of π. We also note that a > 0 for any k.
k
Since χ˜ = π∗χ π∗K is big and nef, Kodaira’s lemma implies there exists an effective
X
divisor D˜ whose su∈pport coincide with the exceptional locus E,F and
χ˜ ε[D˜] is Ka¨hler,
−
hence there exists a hermitian metric h on the line bundle associated to D such that
D˜
χ˜ εRic(h ) > 0.
− D˜
We write D˜ = D˜′ + D˜′′, where suppD′′ = E, and E D˜′. Let σ , σ , σ be the defining
6⊂ E F D˜
sectionofE,F andD˜, respectively. Herethesesectionsaremulti-valuedholomorphicsections
8 BIN GUO
which become global after taking some power. There also exist hermtian metrics h , h , and
E F
h such that
D˜
π∗Ω = σ 2(n−1) σ 2 Ω˜,
| E|hE | F|hF
for some smooth volume form Ω˜ on X˜.
We fix a Ka¨hler metric ω˜ on X˜. The Ka¨hler Ricci flow on X is pulled back to X˜ by the
map π, and it safeties the equation
∂ (χ˜+e−t(π∗ω χ˜)+i∂∂¯π∗ϕ)n
(4.1) π∗ϕ = log 0 − π∗ϕ,
∂t σ 2(n−1) σ 2 Ω˜ −
| E|hE | F|hF
with the initial π∗ϕ(0) = 0. By the previous estimates, we see that π∗ϕ satisfies the estimates
δlog σ 2 C π∗ϕ(t) C, t 0 and δ (0,1).
| D˜|hD˜ − δ ≤ ≤ ∀ ≥ ∀ ∈
We will consider a family of perturbed parabolic Monge-Ampere equations for ǫ (0,1)
∈
∂ (χ˜+e−t(π∗ω χ˜)+ǫω˜ +i∂∂¯π∗ϕ)n
0
ϕ˜ = log − ϕ˜ ,
(4.2)  ∂t ǫ ( σ 2(n−1) +ǫ)( σ 2 +ǫ)Ω˜ − ǫ
 | E|hE | F|hF
ϕ˜ (0) = 0
ǫ


where ϕ˜ (t) PSH(X˜,χ˜+e−t(π∗ω χ˜)+ǫω˜). The equation (4.2) has long time existence
ǫ 0
∈ −
[27], and we will show that solutions to (4.2) converge to that of (4.1) in some sense.
It’s easy to check that the Ka¨hler metrics ω˜ = χ˜ +e−t(π∗ω χ˜) +ǫω˜ + i∂∂¯ϕ˜ satisfies
ǫ 0 ǫ
−
the following evolution equation
∂
(4.3) ω˜ = Ric(ω˜ ) ω˜ +χ˜+ǫω˜ i∂∂¯log ( σ 2(n−1) +ǫ)( σ 2 +ǫ)Ω˜ .
∂t ǫ − ǫ − ǫ − (cid:16) | E|hE | F|hF (cid:17)
By direct calculations, for any smooth nonnegative function f, we have
¯ ¯
i∂∂f ∂f ∂f
¯
i∂∂log(ǫ+f) = ∧
ǫ+f − (f +ǫ)2
f ǫ
¯ ¯
= i∂∂logf + ∂f ∂f
f +ǫ f(f +ǫ)2 ∧
f
¯
i∂∂logf,
≥ f +ǫ
in the smooth sense on X˜ f = 0 and globally as currents. So
\{ }
i∂∂¯log ( σ 2(n−1) +ǫ)( σ 2 +ǫ)Ω˜
(cid:16) | E|hE | F|hF (cid:17)
(n 1) σ 2(n−1) σ 2
− | E|hE Ric(h ) | F|hF Ric(h ) Ric(Ω˜)
≥ − |σE|2h(En−1) +ǫ E − |σF|2hF +ǫ F −
Cω˜,
≥ −
for some uniform constant C independent of ǫ. Thus away from suppD˜ = suppE suppF,
∪
we have
∂
(4.4) ω˜ Ric(ω˜ ) ω˜ +Cω˜.
ǫ ǫ ǫ
∂t ≤ − −
ON THE KA¨HLER RICCI FLOW ON PROJECTIVE MANIFOLDS OF GENERAL TYPE 9
Lemma 4.1. Let ϕ˜ be the solution to (4.2), then there exists a constant C > 0 such that
ǫ
for any t 0, ǫ (0,1), we have
≥ ∈
∂ϕ˜
ǫ
supϕ˜ (t, ) C, sup (t, ) C.
ǫ
· ≤ ∂t · ≤
X˜ X˜
Proof. Let
V = ( σ 2(n−1) +ǫ)( σ 2 +ǫ)Ω˜
ǫ ZX˜ | E|hE | F|hF
be the volume with respect to the volume form ( σ 2(n−1) +ǫ)( σ 2 +ǫ)Ω˜. We see that
| E|hE | F|hF
V V V = Ω,
1 ≥ ǫ ≥ 0 Z
X
hence V is uniformly bounded. We consider (for simplicity we denote Ω˜ = ( σ 2(n−1) +
ǫ ǫ | E|hE
ǫ)( σ 2 +ǫ)Ω˜)
| F|hF
∂ 1
ϕ˜ Ω˜
∂t(cid:16)Vǫ ZX˜ ǫ ǫ(cid:17)
1 (χ˜+e−t(π∗ω χ˜)+ǫω˜ +i∂∂¯ϕ˜ )n 1
= log 0 − ǫ Ω˜ ϕ˜ Ω˜
Vǫ ZX˜ (|σE|2h(En−1) +ǫ)(|σF|2hF +ǫ)Ω˜ ǫ − Vǫ ZX˜ ǫ ǫ
1
log (χ˜+e−t(π∗ω χ˜)+ǫω˜ +i∂∂¯ϕ˜ )n ϕ˜ Ω˜
≤ (cid:16)ZX˜ 0 − ǫ (cid:17)− Vǫ ZX˜ ǫ ǫ
1
C ϕ˜ Ω˜ ,
≤ − V Z ǫ ǫ
ǫ X˜
where for the first inequality we use Jensen’s inequality. From the above we see that
1
ϕ˜ Ω˜ C.
V Z ǫ ǫ ≤
ǫ X˜
Since ϕ˜ PSH(X˜,χ˜+e−t(π∗ω χ˜)+ǫω˜), the mean value inequality implies the uniform
ǫ 0
∈ −
upper bound of ϕ˜
ǫ
supϕ˜ (t) C.
ǫ
≤
X˜
Direct calculations show that (we will denote ϕ˜˙ = ∂ ϕ˜ )
ǫ ∂t ǫ
∂
ϕ˜˙ = ∆ ϕ˜˙ tr e−t(π∗ω χ˜) ϕ˜˙
∂t ǫ ω˜ǫ ǫ − ω˜ǫ 0 − − ǫ
= ∆ ϕ˜˙ n+tr χ˜+ǫtr ω˜ +∆ ϕ˜ ϕ˜˙ .
ω˜ǫ ǫ − ω˜ǫ ω˜ǫ ω˜ǫ ǫ − ǫ
Hence
∂
(4.5) ( ∆ ) (et 1)ϕ˜˙ ϕ˜ = tr π∗ω +n ǫtr ω˜ n,
∂t − ω˜ǫ (cid:16) − ǫ − ǫ(cid:17) − ω˜ǫ 0 − ω˜ǫ ≤
then maximum principle implies
ϕ˜ +nt
(4.6) ϕ˜˙ (t) ǫ C, t > 0.
ǫ ≤ et 1 ≤ ∀
−
(cid:3)
10 BIN GUO
Lemma 4.2. For any δ (0,1), there is a constant C = C such that for any t 0, we have
δ
∈ ≥
ϕ˜ (t) δlog σ 2 C .
ǫ ≥ | D˜|hD˜ − δ
Proof. We will apply the maximum principle. For any small δ > 0 such that
χ˜ δRic(h ) > 0,
− D˜
where we may also assume σ 2 1 by rescaling the metric h . We consider the function
| D˜|hD˜ ≤ D˜
H := ϕ˜ δlog σ 2 . On X˜ D˜, H satisfies the equation
ǫ − | D˜|hD˜ \
∂H (χ˜+e−t(π∗ω χ˜)+ǫω˜ δRic(h )+i∂∂¯H)n
(4.7) = log 0 − − D˜ H δlog σ 2 .
∂t (|σE|2(n−1)hE +ǫ)(|σF|2hF +ǫ)Ω˜ − − | D˜|hD˜
By choosing δ even smaller, we may obtain ω′ := χ˜ δRic(h )+e−t(π∗ω χ˜)+ǫω˜ is a
ǫ − D˜ 0 −
Ka¨hler metric on X˜ for all t 0, and these metrics are uniformly equivalent to ω˜, i.e., there
≥
exists C > 0 such that
0
C−1ω˜ ω′ C ω˜.
0 ǫ ≤ ǫ ≤ 0
Consider the Monge-Ampere equations
(4.8) (χ˜ δRic(h )+e−t(π∗ω χ˜)+ǫω˜ +i∂∂¯ψ )n = eψǫ( σ 2(n−1) +ǫ)( σ 2 +ǫ)Ω˜,
− D˜ 0 − ǫ | E|hE | F|hF
where ψ PSH(X˜,ω′). By the Aubin-Yau theorem, (4.8) admits a unique smooth solution
ǫ ∈ ǫ
ψ for any ǫ (0,1) and t 0. It can be seen that
ǫ
∈ ≥
1 1
eψǫΩ˜ (ω′)n C.
V Z ǫ ≤ V Z ǫ ≤
ǫ X˜ ǫ X˜
Hence mean value inequality implies sup ψ C. Then by [9], we have inf ψ C,
X˜ ǫ ≤ X˜ ǫ ≥ −
hence
ψǫ L∞ C,
k k ≤
for some C independent of ǫ (0,1) and t 0.
Denote ω′(t) = ω′ +i∂∂¯ψ ∈. Taking deriv≥ative with respective to t on both sides of (4.8),
ǫ ǫ ǫ
we get
(4.9) ∆ω′(t)ψ˙ǫ trω′(t)e−t(π∗ω0 χ˜) = ψ˙ǫ.
ǫ − ǫ −
∆ωǫ′(t)ψǫ = n−trωǫ′(t)(cid:16)χ˜−δRic(hD˜)+e−t(π∗ω0 −χ˜)+ǫω˜(cid:17),
Hence
∆ωǫ′(t)(ψ˙ǫ −ψǫ) = ψ˙ǫ −n+trωǫ′(t)(cid:16)χ˜−δRic(hD˜)+2e−t(π∗ω0 −χ˜)+ǫω˜(cid:17)
Noting that χ˜ δRic(h )+2e−t(π∗ω χ˜)+ ǫω˜ > 0 is δ is chosen appropriately, hence at
− D˜ 0 −
˙ ˙
the maximum point of ψ ψ , we have ψ n, thus
ǫ ǫ ǫ
− ≤
˙
ψ C +n C.
ǫ
≤ ≤
Let G = H ψ = ϕ˜ δlog σ 2 ψ . On X˜ D˜ it satisfies the equation
− ǫ ǫ − | D˜|hD˜ − ǫ \
(4.10)
∂G (χ˜ δRic(h )+e−t(π∗ω χ˜)+ǫω˜ +i∂∂¯ψ +i∂∂¯G)n
= log − D˜ 0 − ǫ G δlog σ 2 ψ˙ .
∂t (χ˜−δRic(hD˜)+e−t(π∗ω0 −χ˜)+ǫω˜ +i∂∂¯ψǫ)n − − | D˜|hD˜ − ǫ