84(2010) 427-453

ADIABATIC LIMITS OF RICCI-FLAT K ¨AHLER METRICS

Valentino Tosatti

Abstract

We study adiabatic limits of Ricci-ﬂat K¨ahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic ﬁbration when the volume of the ﬁbers goes to zero. By es- tablishing some new a priori estimates for the relevant complex Monge-Amp`ere equation, we show that the Ricci-ﬂat metrics col- lapse (away from the singular ﬁbers) to a metric on the base of the ﬁbration. This metric has Ricci curvature equal to a Weil- Petersson metric that measures the variation of complex structure of the Calabi-Yau ﬁbers. This generalizes results of Gross-Wilson forK3 surfaces to higher dimensions.

1. Introduction

In this paper, which is a continuation of [To2], we study the behaviour of Ricci-ﬂat K¨ahler metrics on a compact Calabi-Yau manifold when the K¨ahler class degenerates to the boundary of the K¨ahler cone. Given a compact Calabi-Yau manifold, a fundamental theorem of Yau [Y1]

says that there exists a unique Ricci-ﬂat K¨ahler metric in each K¨ahler class. If we now move the K¨ahler class inside the K¨ahler cone, the corresponding Ricci-ﬂat metrics vary smoothly, as long as the class does not approach the boundary of the K¨ahler cone. The question that we want to address is to understand what happens to the Ricci-ﬂat metrics if the class goes to the boundary of the K¨ahler cone. This question has been raised by Yau and reiterated by others, see for example [Y3, W, McM]. In our previous work [To2] we have studied the case when the limit class has positive volume. In this paper we consider the case when the limit volume is zero, and we focus on the situation when the Calabi- Yau manifold admits a holomorphic ﬁbration to a lower dimensional space such that the limit class is the pullback of a K¨ahler class from the base.

To state our result, let us introduce some notation. Let (X, ω_{X})
be a compact K¨ahler n-manifold with c_{1}(X) = 0 in H^{2}(X,R). The
condition that c1(X) = 0 is equivalent to the requirement that the

Received 05/29/2009.

427

canonical bundle ofXbe torsion (see [To2]), and we callXa Calabi-Yau
manifold. We assume that there is a holomorphic mapf :X →Z where
(Z, ω_{Z}) is another compact K¨ahler manifold. We denote byY the image
ofXunderf, and we assume thatY is an irreducible normal subvariety
of Z of dimensionm with 0 < m < n, and that the map f : X → Y
has connected ﬁbers. Then ω_{0} = f^{∗}ω_{Z} is a smooth nonnegative (1,1)
form onX, whose cohomology class lies on the boundary of the K¨ahler
cone. We will also denote by ω_{Y} the restriction of ω_{Z} to the regular
part of Y. The map f : X → Y is an “algebraic ﬁber space” in the
sense of [La] and we can ﬁnd a subvariety S ⊂X such that Y\f(S) is
smooth and f :X\S → Y\f(S) is a smooth submersion (S consists of
singular ﬁbers, as well as ﬁbers of dimension strictly larger thann−m).

Then for any y ∈Y\f(S) the ﬁber X_{y} =f^{−}^{1}(y) is a smooth (n−m)-
manifold, equipped with the K¨ahler form ωX|Xy. Notice that since f
is a submersion near X_{y}, we have thatc_{1}(X_{y}) =c_{1}(X)|Xy, and so the
ﬁbers Xy with y ∈Y\f(S) are themselves Calabi-Yau. Yau’s theorem
[Y1] says that in each K¨ahler class ofX there is a unique K¨ahler metric
with Ricci curvature identically zero. For each 0< t≤1 we call ˜ωt the
Ricci-ﬂat K¨ahler metric cohomologous to [ω_{0}] +t[ω_{X}], and we wish to
study the behaviour of these metrics whentgoes to zero. First of all in
[To2] we proved the following

Theorem 1.1 (Theorem 3.1 of [To2]). The Ricci-flat metricsω˜t on X have uniformly bounded diameter as tgoes to zero.

The volume of any ﬁberX_{y} with respect to ˜ω_{t}is comparable tot^{n}^{−}^{m},
and the Ricci-ﬂat metrics ˜ω_{t} approach an “adiabatic limit”. We will
show that the metrics ˜ω_{t} collapse to a K¨ahler metric on Y\f(S).

Our main theorem is the following:

Theorem 1.2. There is a smooth K¨ahler metric ω on Y\f(S) such
that the Ricci-flat metrics ω˜t when t approaches zero converge to f^{∗}ω
weakly as currents and also in theC_{loc}^{1,β}topology of potentials on compact
sets of X\S, for any 0< β <1. The metric ω satisfies

(1.1) Ric(ω) =ω_{W P},

on Y\f(S), where ω_{W P} is a Weil-Petersson metric measuring the
change of complex structures of the fibers. Moreover for anyy ∈Y\f(S)
if we restrict to X_{y}, the metrics ω˜_{t} converge to zero in the C^{1} topology
of metrics, uniformly as y varies in a compact set of Y\f(S).

This result generalizes work of Gross and Wilson [GW], who consid-
ered the case whenf :X →Y =P^{1} is an elliptically ﬁberedK3 surface
with 24 singular ﬁbers of type I_{1} (see section 5). They achieved their
result by writing down explicit approximations of the Ricci-ﬂat metrics

near the adiabatic limit. A similar approach in higher dimensions seems out of reach at present. Instead, our main technical tool are some new generala priori estimates for complex Monge-Amp`ere equations on the total space of a holomorphic ﬁbration.

The limit equation (1.1) has ﬁrst been explicitly proposed by Song- Tian [ST1], where ω is called ageneralized K¨ahler-Einstein metric.

Let us brieﬂy explain the meaning of ωW P, referring the reader to
section 4 for more details. We have already remarked that the smooth
ﬁbers X_{y} of f are themselves Calabi-Yau (n−m)-manifolds, polarized
by ω_{X}|Xy. If the canonical bundles of the ﬁbers are actually trivial
we get a map from Y\f(S) to the moduli space of polarized Calabi-
Yau manifolds and by pulling back the Weil-Petersson metric we get
a smooth nonnegative form ω_{W P} on Y\f(S) (a similar construction
goes through in the case when the ﬁbers have torsion canonical bundle).

Notice thatω_{W P} is identically zero precisely when the complex structure
of the ﬁbers doesn’t change. The appearance of the Weil-Petersson
metric in the more general setting of adiabatic limits of constant scalar
curvature K¨ahler metrics was observed by Song-Tian [ST1], and further
studied by Fine [Fi] and Stoppa [St].

There are two possible situations that we have in mind for our setup:

in one caseY is smooth, and then we can just takeZ =Y. In the second
case we take Z = P^{N} withω_{Z} the Fubini-Study metric, and then Y is
an algebraic variety. A natural class of examples where this situation
arises is the following: X is a projective Calabi-Yau manifold and L
is a semiample line bundle over X with Iitaka dimension κ(X, L) =
m < n. Then a classical construction of Iitaka (see 2.1.27 in [La])
gives a holomorphic map f : X → P^{N} exactly as in the setup. Note
that if the log Abundance Conjecture holds then every line bundle L
with cohomology class on the boundary of the K¨ahler cone and with
κ(X, L) = m < n is automatically semiample (see [To2]). This is
known to hold ifn= 2.

The organization of the paper is the following. In section 2 we set up
the problem as a family of degenerating complex Monge-Amp`ere equa-
tions and state our estimates that imply the main result. In section 3 we
provea priori C^{2} estimates for these equations as well as C^{3} estimates
along the ﬁbers. In section 4 we use the estimates to prove our main
theorem, and in section 5 we provide a few examples.

Acknowledgments.I would like to thank my advisor Shing-Tung Yau for suggesting this problem and for constant support. I also thank Chen- Yu Chi, Jian Song, G´abor Sz´ekelyhidi and Ben Weinkove for very useful discussions. I was partially supported by a Harvard Merit Fellowship.

These results are part of my PhD thesis at Harvard University [To3].

2. Complex Monge-Amp`ere equations

In this section we translate our problem to a family of degenerating complex Monge-Amp`ere equations, and we state our estimates.

First of all let us recall our setup from the introduction: (X, ω_{X}) is
a compact K¨ahler manifold of complex dimension n with c_{1}(X) = 0,
or in other words a Calabi-Yau manifold. We have a map holomorphic
map f : X → Z, where (Z, ω_{Z}) is another compact K¨ahler manifold,
with image Y ⊂Z and so that f : X → Y has connected ﬁbers. Y is
assumed to be an irreducible normal subvariety of Z of dimension m
with 0< m < n, and we let ω_{Y} be the restriction of ω_{Z} to the regular
part ofY. We also set ω_{0} =f^{∗}ω_{Z}, which is a smooth nonnegative (1,1)
form on X whose cohomology class lies on the boundary of the K¨ahler
cone. There is a proper subvariety S⊂X such that Y\f(S) is smooth
and f : X\S → Y\f(S) is a smooth submersion. Yau’s theorem [Y1]

says that in each K¨ahler class ofX there is a unique K¨ahler metric with
Ricci curvature identically zero. For each 0< t≤1 we call ˜ω_{t}the Ricci-
ﬂat K¨ahler metric cohomologous to [ω_{0}] +t[ω_{X}], and we wish to study
the behaviour of these metrics when tgoes to zero. On X we have

ω^{k}_{0}∧ω_{X}^{n}^{−}^{k}= 0,
form+ 1≤k≤n, and

(2.1) ω^{m}_{0} ∧ω^{n}_{X}^{−}^{m}=Hω^{n}_{X},

where the smooth non-negative functionH vanishes precisely onS and
is such that H^{−}^{γ} is in L^{1} for some small γ > 0. This is because H is
locally comparable to a sum of squares of holomorphic functions (the
minors of the Jacobian of f). In particular it follows that

Z

X

ω_{0}^{m}∧ω_{X}^{n}^{−}^{m} >0.

For later purposes we need the following construction. Let I be the
ideal sheaf off(S) insideZ. We coverZ by a ﬁnite number of open sets
U_{k} so that on eachU_{k} the idealI is generated by holomorphic functions
h_{k,j}, with 1≤j ≤N_{k}. We then ﬁx η_{k} a partition of unity subordinate
to the covering {U_{k}}and we let

(2.2) σ=X

k,j

η_{k}|h_{k,j}|^{2},

if S6=∅and otherwise we just set σ= 1. Thenσ is a smooth nonnega- tive function onZ with zero locus preciselyf(S) and there is a constant C so that onZ we have

(2.3) σ ≤C, 0≤√

−1∂σ∧∂σ≤Cω_{Z}, −Cω_{Z}≤√

−1∂∂σ ≤Cω_{Z}.

Then for any y∈Y\f(S) we have the inequality

(2.4) σ(y)^{λ} ≤Cinf

Xy

H,

for some constants C, λ, and we are free to enlargeλ if needed. This is
because both of the function H and f^{∗}σ on X are locally comparable
to a sum of squares of holomorphic functions and they both have zero
set equal to S. By taking a log resolution of the ideal sheaf ofS inside
X and we can assume thatS is a divisor with simple normal crossings,
and then the holomorphic functions have well deﬁned vanishing orders
along the irreducible components of S, and (2.4) follows.

In this setting we look at the K¨ahler formsωt=ω0+tωX for 0< t≤1,
which are cohomologous to the Ricci-ﬂat metrics ˜ω_{t}. We then deﬁne a
smooth function E by

Ric(ω_{X}) =√

−1∂∂E, Z

X

e^{E}ω_{X}^{n} =
Z

X

ω_{1}^{n},

which is possible thanks to the∂∂-lemma. Then the equation Ric(˜ωt) = 0 is equivalent to

˜

ω_{t}^{n}=a_{t}e^{E}ω_{X}^{n},
where

a_{t}=
R

Xω^{n}_{t}
R

Xω^{n}_{1}.

Using the∂∂-lemma again, we can ﬁnd smooth functionsϕ_{t}for 0< t≤
1 so that ˜ω_{t}=ω_{t}+√

−1∂∂ϕ_{t}, sup_{X}ϕ_{t}= 0 and we have

(2.5) (ω_{t}+√

−1∂∂ϕ_{t})^{n}=a_{t}e^{E}ω_{X}^{n}.

Notice that as tapproaches zero, the constants a_{t} behave like
(2.6)

n m

R

Xω^{m}_{0} ∧ω^{n}_{X}^{−}^{m}
R

Xω^{n}_{1} t^{n}^{−}^{m}+O t^{n}^{−}^{m+1}
.
We can then write (2.5) as

(2.7) (ω_{t}+√

−1∂∂ϕ_{t})^{n}=c_{t}t^{n}^{−}^{m}e^{E}ω_{X}^{n},

where the constant c_{t} is bounded away from zero and inﬁnity astgoes
to zero. Equation (2.7) has been studied for example in [KT] where a
uniform L^{∞} bound onϕ_{t} was conjectured. When m = 1 such a bound
can be easily proved using the Moser iteration method (see [ST1]). The
bound in the general case was then proved independently by Demailly
and Pali [DP] and by Eyssidieux, Guedj and Zeriahi [EGZ2]:

Theorem 2.1 ([DP,EGZ2]). There is a constant C that depends only on X, E, ωX, ω0 such that for all 0< t≤1 we have

(2.8) kϕ_{t}kL^{∞} ≤C.

Our goal is to show higher order estimates for ϕ_{t} which are uniform
on compact sets of X\S. Notice that since

0<tr_{ω}_{X}ω˜_{t}= tr_{ω}_{X}ω_{t}+ ∆_{ω}_{X}ϕ_{t},

and since tr_{ω}_{X}ω_{t}is uniformly bounded, we always have a uniform lower
bound for ∆_{ω}_{X}ϕ_{t}.

The following are our main results, and together they imply Theorem 1.2.

Theorem 2.2. There are constants A, B, C that depend only on the fixed data, so that on X\S and for any 0< t≤1 we have

(2.9) t

Ce^{Ae}^{Bσ}^{−}^{λ}ω_{X} ≤ω˜_{t}≤Ce^{Ae}^{Bσ}

−λ

ω_{X},

where σ is defined by (2.2). In particular the Laplacian ∆_{ω}_{X}ϕ_{t} is
bounded uniformly on compact sets of X\S, independent of t.

Theorem 2.3. Given anyy∈Y\f(S)denote byX_{y} the fiberf^{−}^{1}(y),
by ω_{y} the K¨ahler form ω_{X}|Xy and by ω˜_{y} the restriction of the Ricci-flat
metricω˜_{t}|Xy. Then there are constants A, B, C that only depend on the
fixed data, so that on the fiber Xy and any 0< t≤1 we have

(2.10) t

Ce^{Ae}^{Bσ(y)}^{−}^{λ}ω_{y}≤ω˜_{y} ≤tCe^{Ae}^{Bσ(y)}

−λ

ω_{y},

(2.11) |∇ω˜y|^{2}ωy ≤t^{1/2}Ce^{Ae}^{Bσ(y)}

−λ

,

where ∇ is the covariant derivative of ωy. In particular the metrics ω˜y

converge to zero in C^{1}(ω_{y}) as t approaches zero, uniformly as y varies
in a compact set of Y\f(S).

Theorem 2.4. As t→ 0 the Ricci-flat metricsω˜_{t} on X\S converge
tof^{∗}ω, whereωis a smooth K¨ahler metric onY\f(S). The convergence
is weakly as currents and also in the C_{loc}^{1,β} topology of K¨ahler potentials
for any 0< β <1. The metricω satisfies

Ric(ω) =ω_{W P},

on Y\f(S), where ω_{W P} is the pullback of the Weil-Petersson metric
from the moduli space of the Calabi-Yau fibers, and it measures the
change of complex structures of the fibers.

3. A priori estimates

In this section we prove a priori C^{2} estimates for the degenerating
complex Monge-Amp`ere equations that we are considering, and we also
proveC^{3} estimates along the ﬁbers off.

We start with a few lemmas.

Lemma 3.1. There is a uniform constant C so that for all0< t≤1 we have

(3.1) tr_{˜}_{ω}_{t}ω_{0}≤C.

Proof. Recall that we are assuming thatω_{0} =f^{∗}ω_{Z} wheref :X→Z
is a holomorphic map. We can then use the Chern-Lu formula that
appears in Yau’s Schwarz lemma computation [Y2,To1] and get

∆_{ω}_{˜}_{t}log tr_{ω}_{˜}_{t}ω_{0} ≥ −Atr_{ω}_{˜}_{t}ω_{0},
for a uniform constant A. Noticing that

∆ω˜tϕt=n−trω˜tωt≤n−trω˜tω0, we see that

(3.2) ∆_{ω}_{˜}_{t}(log tr_{ω}_{˜}_{t}ω_{0}−(A+ 1)ϕ_{t})≥tr_{˜}_{ω}_{t}ω_{0}−n(A+ 1).

Then the maximum principle applied to (3.2), together with the esti-

mate (2.8), gives (3.1). q.e.d.

The next lemma, which gives a Sobolev constant bound, is due inde- pendently to Allard [A] and Michael-Simon [MS].

Lemma 3.2. There is a uniform constant C so that for any 0< t≤
1, for any y∈Y\f(S) and for any u∈C^{∞}(X_{y}) we have

(3.3) Z

Xy

|u|^{2(n}

−m)
n−m−1ω_{y}^{n}^{−}^{m}

!^{n}^{−}_{n}^{m}^{−}^{1}

−m

≤C Z

Xy

|∇u|^{2}ωy+|u|^{2}
ω^{n}_{y}^{−}^{m}.
Proof. For anyy∈Y\f(S) the ﬁberX_{y} is a smooth (n−m) - dimen-
sional complex submanifold ofX. SinceX is K¨ahler, it follows thatX_{y}
is a minimal submanifold, and so it has vanishing mean curvature vec-
tor. We then use the Nash embedding theorem to isometrically embed
(X, ω_{X}) into Euclidean space, and so we have an isometric embedding
X → R^{N}. The length of the mean curvature vector of the composite
isometric embedding X_{y} → X → R^{N} is then uniformly bounded inde-
pendent of y, since it depends only on the second fundamental form of
X → R^{N}. Then (3.3) follows from the uniform Sobolev inequality of
[A,MS]. Notice that they prove anL^{1} Sobolev inequality, but this im-
plies the stated L^{2} Sobolev inequality thanks to the H¨older inequality.

q.e.d.

One can easily avoid the Nash embedding theorem by using a par- tition of unity to reduce directly to the Euclidean case, but the above proof is perhaps cleaner.

We note here that the volume ofX_{y} with respect toω_{y},R

Xyω_{y}^{n}^{−}^{m}, is
a homological constant independent of y ∈ Y\f(S), and up to scaling
ω_{X} we may assume that it is equal to 1. The next step is to prove a
diameter bound for ω_{y}:

Lemma 3.3. There is a uniform constant C so that for any 0< t≤ 1, for any y∈Y\f(S) we have

(3.4) diam(Xy, ωy)≤C.

Proof. As above we embed (X, ωX) isometrically intoR^{N} and we get
that the length of the mean curvature vector of the composite isometric
embeddingX_{y} →X→R^{N} is then uniformly bounded independent ofy.

We can then apply Theorem 1.1 of [Tp] and get the required diameter bound.

Alternatively, ﬁrst one observes that (3.3) implies that there is a
uniform constant κ so that geodesic balls in X_{y} of radius r < 1 have
volume at leastκr^{2(n}^{−}^{m)} (Lemma 3.2 in [H]). Since the total volume of
Xy is constant equal to 1, an elementary argument gives the required

diameter bound. q.e.d.

The next step is to prove a Poincar´e inequality for the restricted met-
ric ω_{y}. This time the constant will not be uniformly bounded, but it
will blow up like a power of _{H}^{1}, whereH is deﬁned in (2.1). To this end,
we ﬁrst estimate the Ricci curvature of ω_{y}. Fix a point y ∈ Y\f(S)
and choose local coordinates z^{1}, . . . , z^{n}^{−}^{m} on the ﬁber X_{y}, which ex-
tend locally to coordinates in a ball in X. Then pick local coordi-
natesw^{n}^{−}^{m+1}, . . . , w^{n}neary∈Y\f(S), so thatz^{1}, . . . , z^{n}^{−}^{m}, z^{n}^{−}^{m+1} =
f^{∗}(w^{n}^{−}^{m+1}), . . . , z^{n}=f^{∗}(w^{n}) give local holomorphic coordinates onX.

We can also assume that at the point y the metric ω_{Y} is the identity.

At any ﬁxed point of X_{y} we then have
Ric(ω_{y}) =−√

−1∂∂log ω_{y}^{n}^{−}^{m}
dz^{1}∧ · · · ∧dz^{n}^{−}^{m}

=−√

−1∂∂log ω_{X}^{n}^{−}^{m}∧ω^{m}_{0}
dz^{1}∧ · · · ∧dz^{n}

=−√

−1∂∂logH−√

−1∂∂log ω_{X}^{n}
dz^{1}∧ · · · ∧dz^{n}

≥ −

√−1∂∂H

H + Ric(ω_{X})|Xy

≥ − C

H +C

ωy≥ −C Hωy, (3.5)

where all derivatives are in ﬁber directions. Combining (3.5) and (2.4)
we see that the Ricci curvature of ω_{y} is bounded below by −Cσ^{−}^{λ}.
Since the diameter of ω_{y} is bounded by Lemma 3.3, a theorem of Li-
Yau [LY] then shows that the Poincar´e constant ofω_{y} is bounded above
by Ce^{Bσ}^{−}^{λ}. This proves the following

Lemma 3.4. There are uniform constantsλ, B, Cso that for any0<

t≤1, for anyy∈Y\f(S)and for anyu∈C^{∞}(X_{y})withR

Xyuω^{n}_{y}^{−}^{m}= 0

we have (3.6)

Z

Xy

|u|^{2}ω^{n}_{y}^{−}^{m} ≤Ce^{Bσ}^{−}^{λ}
Z

Xy

|∇u|^{2}ωyω_{y}^{n}^{−}^{m}.

We now let ˜ω_{y} be the restriction ˜ω_{t}|Xy. We have the following esti-
mate for the volume form of ˜ω_{y} on X_{y}:

˜
ω_{y}^{n}^{−}^{m}
ωy^{n}^{−}^{m}

= ω˜^{n}_{t}^{−}^{m}∧ω_{0}^{m}

ω^{n}_{X}^{−}^{m}∧ω_{0}^{m} = ω˜^{n}_{t}^{−}^{m}∧ω_{0}^{m}

˜

ω_{t}^{n} · ω˜_{t}^{n}
Hω_{X}^{n}

≤

ω˜^{n}_{t}^{−}^{1}∧ω_{0}

˜
ω^{n}_{t}

m

c_{t}t^{n}^{−}^{m}e^{E}
H

= (trω˜tω0)^{m}c_{t}t^{n}^{−}^{m}e^{E}

H ≤ Ct^{n}^{−}^{m}
σ^{λ} .
(3.7)

Notice that when we restrict toX_{y} we have

˜

ω_{y} = (ω_{0}+tω_{X} +√

−1∂∂ϕ_{t})|Xy =tω_{y}+ (√

−1∂∂ϕ_{t})|Xy.
It is convenient to deﬁne a functionϕ_{t} on Y\f(S) by

ϕ_{t}(y) =
Z

Xy

ϕ_{t}ω_{y}^{n}^{−}^{m}.

This is just the “integration along the ﬁbers” of ϕt, and we will also
denote by ϕ_{t} its pullback to X\S via f. We also deﬁne a function on
X\S by

ψ= 1

t ϕ_{t}−ϕ_{t}
,
so that we have R

Xyψω_{y}^{n}^{−}^{m}= 0 and onX_{y} we have

(3.8) (ω_{y}+√

−1∂∂ψ)^{n}^{−}^{m} = ω˜_{y}^{n}^{−}^{m}
t^{n}^{−}^{m} ≤ C

σ^{λ}ω^{n}_{y}^{−}^{m}.

We can then apply Yau’sL^{∞}estimate for complex Monge-Amp`ere equa-
tions [Y1] to the inequality (3.8). Since the volume of X_{y} is constant
equal to 1, the Sobolev constant of ω_{y} is uniformly bounded (Lemma
3.2) and the Poincar´e constant is controlled by Lemma 3.4, Yau’s L^{∞}
estimate gives

(3.9) sup

Xy

ϕ_{t}−ϕ_{t}

=tsup

Xy

|ψ| ≤tCe^{Bσ(y)}^{−}^{λ},

where we increased the constant B to absorb the term σ^{−}^{λ} in (3.8).

Recall that from (2.8) we have a uniform bound for the oscillation of
ϕ_{t}.

Proof of Theorem 2.2. First we will show the right-hand side inequal- ity in (2.9). We will apply the maximum principle to the quantity

K=e^{−}^{Bσ}^{−}^{λ}

log tr_{ω}_{X}ω˜_{t}−A

t(ϕ_{t}−ϕ_{t})

,

where Ais a suitably chosen uniform large constant. The maximum of K on X\S is obviously achieved, and we will show that K ≤ C for a uniform constant C. This together with (3.9) will show that on X\S we have

(3.10) ∆_{ω}_{X}ϕ_{t}= tr_{ω}_{X}ω˜_{t}−tr_{ω}_{X}ω_{0}−nt≤tr_{ω}_{X}ω˜_{t}≤Ce^{Ce}^{Bσ}

−λ

,
which is half of (2.9) To do this, we ﬁrst compute as in Yau’s C^{2} esti-
mates [Y1]

∆_{ω}_{˜}_{t}log tr_{ω}_{X}ω˜_{t}≥ −Ctr_{ω}_{˜}_{t}ω_{X} −C,
for a uniform constant C. On the other hand

∆_{ω}_{˜}_{t}ϕ_{t}≤n−t·tr_{ω}_{˜}_{t}ω_{X},
and so if A is large enough we get

∆_{ω}_{˜}_{t}

log tr_{ω}_{X}ω˜_{t}− A
tϕ_{t}

≥tr_{˜}_{ω}_{t}ω_{X} −C
t.

Since f is locally a submersion onX\S, the ﬁber integration formula

∂∂ϕ_{t} =f∗(∂∂ϕ_{t}∧ω^{n}_{X}^{−}^{m})
holds. So we can compute that

∆_{ω}_{˜}_{t}ϕ_{t}= tr_{ω}_{˜}_{t}f∗(√

−1∂∂ϕ_{t}∧ω_{X}^{n}^{−}^{m})

= tr_{ω}_{˜}_{t}f∗((˜ω_{t}−ω_{t})∧ω^{n}_{X}^{−}^{m})

≥ −tr_{ω}_{˜}_{t}f∗(ω_{t}∧ω^{n}_{X}^{−}^{m})

=−tr_{ω}_{˜}_{t}f∗(f^{∗}ω_{Y} ∧ω_{X}^{n}^{−}^{m})−ttr_{ω}_{˜}_{t}f∗(ω^{n}_{X}^{−}^{m+1})

=−tr_{ω}_{˜}_{t}ω_{0}−ttr_{˜}_{ω}_{t}f∗(ω^{n}_{X}^{−}^{m+1}).

(3.11)

On Y\f(S) the K¨ahler formf∗(ω^{n}_{X}^{−}^{m+1}) can be estimated by

f∗(ω_{X}^{n}^{−}^{m+1})≤ ω_{Y}^{m}^{−}^{1}∧f∗(ω^{n}_{X}^{−}^{m+1})

ω_{Y}^{m} ω_{Y} = f∗(ω^{m}_{0}^{−}^{1}∧ω^{n}_{X}^{−}^{m+1})
ω^{m}_{Y} ω_{Y}

≤Cf∗(ω_{X}^{n})

ω_{Y}^{m} ωY =Cf∗(H^{−}^{1}ω^{m}_{0} ∧ω^{n}_{X}^{−}^{m})
ω_{Y}^{m} ωY

≤Cσ^{−}^{λ}f∗(ω_{0}^{m}∧ω_{X}^{n}^{−}^{m})

ω^{m}_{Y} ω_{Y} =Cσ^{−}^{λ}ω_{Y}.
(3.12)

and so using (3.1) we get

∆_{ω}_{˜}_{t}ϕ_{t}≥ −C−tCσ^{−}^{λ}.
It follows that

(3.13) ∆_{ω}_{˜}_{t}

log tr_{ω}_{X}ω˜_{t}−A

t(ϕ_{t}−ϕ_{t})

≥tr_{ω}_{˜}_{t}ω_{X} −C

t −Cσ^{−}^{λ}.

Using (2.3) and (3.1) we have that

(3.14) |∆ω˜tσ| ≤Ctrω˜tω0 ≤C,
(3.15) |∇σ|^{2}ω˜t ≤Ctr_{ω}_{˜}_{t}ω_{0} ≤C,
Using (3.13) we then compute

∆ω˜tK≥e^{−}^{Bσ}^{−}^{λ}

tr˜ωtωX −C

t −Cσ^{−}^{λ}

+

log tr_{ω}_{X}ω˜_{t}−A

t(ϕ_{t}−ϕ_{t})

∆_{ω}_{˜}_{t}

e^{−}^{Bσ}^{−}^{λ}
+ 2e^{Bσ}^{−}^{λ}Reh∇K,∇e^{−}^{Bσ}^{−}^{λ}iω˜t

−2

log tr_{ω}_{X}ω˜_{t}−A

t(ϕ_{t}−ϕ_{t})

e^{Bσ}^{−}^{λ}

∇e^{−}^{Bσ}^{−}^{λ}

2

˜ ωt

. (3.16)

Using (2.3), (3.14) and (3.15), the second term in (3.16) can be esti- mated as follows

∆_{ω}_{˜}_{t}

e^{−}^{Bσ}^{−}^{λ}

= Bλe^{−}^{Bσ}^{−}^{λ}

σ^{λ+1} ∆_{ω}_{˜}_{t}σ+B^{2}λ^{2}e^{−}^{Bσ}^{−}^{λ}
σ^{2λ+2} |∇σ|^{2}ω˜t

−Bλ(λ+ 1)e^{−}^{Bσ}^{−}^{λ}
σ^{λ+2} |∇σ|^{2}ω˜t

≥ −Ce^{−}^{Bσ}^{−}^{λ}

σ^{λ+1} −Ce^{−}^{Bσ}^{−}^{λ}
σ^{λ+2}

≥ −Ce^{−}^{Bσ}^{−}^{λ}
σ^{λ+2} .
(3.17)

At the maximum of K we may assume that K ≥0, otherwise we have nothing to prove. Hence we can use (3.9) to estimate

log tr_{ω}_{X}ω˜_{t}−A

t(ϕ_{t}−ϕ_{t})

∆_{ω}_{˜}_{t}

e^{−}^{Bσ}^{−}^{λ}

≥ −Ce^{−}^{Bσ}^{−}^{λ}

σ^{λ+2} log tr_{ω}_{X}ω˜_{t}− C
σ^{λ+2}.
(3.18)

The fourth term in (3.16) can be estimated using (3.15)

∇e^{−}^{Bσ}^{−}^{λ}

2

˜ ωt

= B^{2}λ^{2}e^{−}^{2Bσ}^{−}^{λ}

σ^{2λ+2} |∇σ|^{2}ω˜t ≤ Ce^{−}^{2Bσ}^{−}^{λ}
σ^{2λ+2} ,
(3.19)

−2

log tr_{ω}_{X}ω˜_{t}−A

t(ϕ_{t}−ϕ_{t})

e^{Bσ}^{−}^{λ}

∇e^{−}^{Bσ}^{−}^{λ}

2

˜ ωt

≥ −Ce^{−}^{Bσ}^{−}^{λ}

σ^{2λ+2} log tr_{ω}_{X}ω˜_{t}− C
σ^{2λ+2}.
(3.20)

Plugging (3.18) and (3.20) in (3.16), at the maximum point ofK we get
0≥tr_{ω}_{˜}_{t}ω_{X}−C

t − C

σ^{λ} − C

σ^{2λ+2} log tr_{ω}_{X}ω˜_{t}−Ce^{Bσ}^{−}^{λ}
σ^{2λ+2}.

Since for any two K¨ahler metricsω,ω˜ we have
(3.21) tr_{ω}ω˜ ≤(tr_{ω}_{˜}ω)^{n}^{−}^{1}ω˜^{n}

ω^{n},
we see that

tr_{ω}_{X}ω˜_{t}≤Ct^{n}^{−}^{m}(tr_{ω}_{˜}_{t}ω_{X})^{n}^{−}^{1} ≤C(tr_{ω}_{˜}_{t}ω_{X})^{n}^{−}^{1},

and using this and the inequalities 2ab≤εa^{2}+b^{2}/εand (logx)^{2} ≤x+C
we get

tr_{ω}_{˜}_{t}ω_{X} ≤ C
t + C

σ^{λ} + C

σ^{4λ+4} +Ce^{Bσ}^{−}^{λ}
σ^{2λ+2} +1

2tr_{ω}_{˜}_{t}ω_{X},
whence

tr_{ω}_{˜}_{t}ω_{X} ≤ C

t +Ce^{Cσ}^{−}^{λ}.
At the same point we then get

tr_{˜}_{ω}_{t}ω_{t}= tr_{ω}_{˜}_{t}(ω_{0}+tω_{X})≤C+tCe^{Cσ}^{−}^{λ}.
and using (3.21) we get

(3.22) tr_{ω}_{t}ω˜_{t}≤(tr_{ω}_{˜}_{t}ω_{t})^{n}^{−}^{1}ω˜_{t}^{n}
ω_{t}^{n} ≤

C+tCe^{Cσ}^{−}^{λ}n−1ω˜_{t}^{n}
ω_{t}^{n}.
We now use (2.1), (2.7) and (2.4) to get

(3.23) ω˜_{t}^{n}

ω_{t}^{n} ≤ Ct^{n}^{−}^{m}ω_{X}^{n}

ω^{m}_{0} ∧(tω_{X})^{n}^{−}^{m} = C
H ≤ C

σ^{λ}.
Combining (3.22) and (3.23) we get

tr_{ω}_{t}ω˜_{t}≤Ce^{Cσ}^{−}^{λ},

for some uniform constant C. But we also have ω_{t}=ω_{0}+tω_{X} ≤Cω_{X}
and so we get

tr_{ω}_{X}ω˜_{t}≤Ce^{Cσ}^{−}^{λ}.

Using (3.9) again, this implies that at the maximum of K we have
K ≤C+e^{−}^{Bσ}^{−}^{λ}log

Ce^{Cσ}^{−}^{λ}

≤C.

We now show the left-hand side inequality in (2.9). To this extent we apply the maximum principle to the quantity

K_{1}=e^{−}^{Bσ}^{−}^{h}^{λ}

log(t·tr_{ω}_{˜}_{t}ω_{X})−A

t(ϕ_{t}−ϕ_{t})

,

where Ais a suitably chosen uniform large constant. The maximum of K1 on X\S is obviously achieved, and we will show that K1 ≤ C for a uniform constant C. This together with (3.9) will show that on X\S we have

(3.24) tr_{ω}_{˜}_{t}ω_{X} ≤ C

te^{Ce}^{Bσ}

−λ

,

which is the other half of (2.9). To prove that K_{1} ≤ C we use the
maximum principle and, as in (3.16), we compute

∆_{ω}_{˜}_{t}K_{1} ≥e^{−}^{Bσ}^{−}^{λ}

tr_{ω}_{˜}_{t}ω_{X} −C

t −Cσ^{−}^{λ}

+

log(t·tr_{˜}_{ω}_{t}ω_{X})− A

t(ϕ_{t}−ϕ_{t})

∆_{ω}_{˜}_{t}

e^{−}^{Bσ}^{−}^{λ}
+ 2e^{Bσ}^{−}^{λ}Reh∇K_{1},∇e^{−}^{Bσ}^{−}^{λ}iω˜t

−2

log(t·tr_{ω}_{˜}_{t}ω_{X})− A

t(ϕ_{t}−ϕ_{t})

e^{Bσ}^{−}^{λ}

∇e^{−}^{Bσ}^{−}^{λ}

2

˜ ωt

. (3.25)

We estimate this in the same way as before and get

∆˜ωtK1 ≥e^{−}^{Bσ}^{−}^{λ}

trω˜tωX−C

t −Cσ^{−}^{λ}

−Ce^{−}^{Bσ}^{−}^{λ}

σ^{2λ+2} log(t·tr_{˜}_{ω}_{t}ω_{X})− C
σ^{2λ+2}
+ 2e^{Bσ}^{−}^{λ}Reh∇K_{1},∇e^{−}^{Bσ}^{−}^{λ}iω˜t.
(3.26)

At the maximum of K_{1} we get
0≥tr_{ω}_{˜}_{t}ω_{X} −C

t − C

σ^{λ} − C

σ^{2λ+2}log(t·tr_{ω}_{˜}_{t}ω_{X})−Ce^{Cσ}^{−}^{λ},

and using the inequalities 2ab≤εa^{2}+b^{2}/εand (logx)^{2} ≤x+C we get
tr_{ω}_{˜}_{t}ω_{X} ≤ C

t + C

σ^{λ} + C

σ^{4λ+4} +Ce^{Cσ}^{−}^{λ}+1

2tr_{ω}_{˜}_{t}ω_{X},
whence

t·tr_{ω}_{˜}_{t}ω_{X} ≤C+tCe^{Cσ}^{−}^{λ} ≤Ce^{Cσ}^{−}^{λ},
and so at that point

K_{1} ≤C+e^{−}^{Bσ}^{−}^{λ}log

Ce^{Cσ}^{−}^{λ}

≤C,

and we are done. q.e.d.

Proof of Theorem 2.3. We will ﬁrst show (2.10), which is an easy consequence of (2.9). The left-hand side follows immediately from (2.9), which implies

(3.27) trω˜yωy ≤ C
t e^{Ce}^{Bσ}

−λ

. Then (3.21) and (3.7) give

(3.28) tr_{ω}_{y}ω˜_{y} ≤(tr_{ω}_{˜}_{y}ω_{y})^{n}^{−}^{m}^{−}^{1}ω˜^{n}_{y}^{−}^{m}

ω^{n}_{y}^{−}^{m} ≤tCe^{Ce}^{Bσ}

−λ

σ^{λ} ≤tCe^{Ce}^{Bσ}

−λ

, which proves (2.10).

Next, we show (2.11). Recall from (3.10) and (3.24) that onX\S we have

(3.29) tr_{ω}_{X}ω˜_{t}≤Ce^{C}^{0}^{e}^{Bσ}

−λ

.
(3.30) tr_{ω}_{˜}_{t}ω_{X} ≤ C

te^{C}^{0}^{e}^{Bσ}

−λ

,

for uniform constantsB, C, C_{0}. We apply the maximum principle to the
quantity

K_{2} =e^{−}^{Ae}^{Bσ}

−λ

S+Ce^{3C}^{0}^{e}^{Bσ}

−λ

t^{5/2} tr_{ω}_{X}ω˜_{t}

! ,

for suitable constants A, C, where the quantity S is the same quantity as in [Y1]:

S =|∇ω˜_{t}|^{2}ω˜t,

where∇is the covariant derivative associated to the metric ω_{X}. Using
ϕ_{t} we can write

S = ˜g^{ip}_{t} ˜g^{qj}_{t} g˜^{kr}_{t}

g_{ij,k}^{0} +ϕ_{ijk}

g_{qp,r}^{0} +ϕ_{pqr}
,

where again lower indices are covariant derivatives with respect to ω_{X},
and where g^{0}_{ij} are the components of ω0. We are going to show that
K_{2} ≤ _{t}5/2^{C} , and using (3.29) this implies that

(3.31) S ≤ Ce^{Ae}^{Bσ}

−λ

t^{5/2} .

We now use (3.28), which says that onXy we have
(3.32) tr_{ω}_{y}ω˜_{y} ≤tCe^{C}^{0}^{e}^{Bσ}

−λ

.

At any given point of X_{y} we can assume thatω_{X} is the identity and ˜ω_{t}
is diagonal with positive entries λ_{i}, 1 ≤i≤n, so that the ﬁrst n−m
directions are tangent to the ﬁber X_{y}. Then (3.32) gives that

(3.33) λ_{i}≤tCe^{C}^{0}^{e}^{Bσ}

−λ

, for 1≤i≤n−m. Then using (3.31) we see that

n−m

X

i,j,k=1

1

λ_{i}λ_{j}λ_{k}|ϕ_{ijk}|^{2} ≤

n

X

i,j,k=1

1
λ_{i}λ_{j}λ_{k}

g_{ij,k}^{0} +ϕ_{ijk}

2 =S ≤ Ce^{Ae}^{Bσ}

−λ

t^{5/2} ,
where we have used thatg^{0}

ij,k vanishes whenever 1≤i, j≤n−m since
ω_{0}|Xy = 0. Using (3.33) we get

|∇ω˜_{y}|^{2}ωy =

n−m

X

i,j,k=1

|ϕ_{ijk}|^{2} ≤t^{1/2}Ce^{(A+3C}^{0}^{)e}^{Bσ}

−λ

, and this is (2.11).

We now prove that K_{2} ≤ _{t}5/2^{C} . To simplify the computation, we will
use the notation

F(x) =e^{xe}^{Bσ}

−λ

,

where x is a real number, and we note here thatF is increasing. The
starting point is a precise formula for ∆˜ωtS. This is just Yau’s C^{3}
estimate [Y1], but without assuming that the metrics ˜ω_{t} and ω_{t} =
ω0 +tωX are equivalent, and it is done in a more general setting in
[TWY] (see also [PSS]). With the notation of [TWY] we can write

S =X

i,j,k

|a^{i}_{jk}|^{2}.

We then choose local unitary frames{θ^{1}, . . . , θ^{n}}forω_{X} and{θ˜^{1}, . . . ,θ˜^{n}}
for ˜ω_{t}, and write

θ˜^{i} =X

j

a^{i}_{j}θ^{j},

θ^{i} =X

j

b^{i}_{j}θ˜^{j},

for some local matrices of functions a^{i}_{j}, b^{i}_{j}. Notice that at any given
point we can choose the frames and arrange that

(3.34) a^{i}_{j} =p

λ_{i}δ_{j}^{i},

(3.35) b^{i}_{j} = 1

√λ_{i}δ^{i}_{j}.
Then in our case [TWY, (4.3)] reads

∆ω˜tS ≥ 2Re

a^{i}_{kℓ}

b^{m}_{k}b^{q}_{ℓ}b^{s}_{p}R_{mqs}^{j} a^{i}_{rp}a^{r}_{j}−a^{i}_{j}b^{q}_{ℓ}b^{s}_{p}R_{mqs}^{j} a^{r}_{kp}b^{m}_{r}

−a^{i}_{j}b^{m}_{k}b^{s}_{p}R^{j}_{mqs}a^{r}_{ℓp}b^{q}_{r}+a^{i}_{j}b^{m}_{k}b^{q}_{ℓ}b^{s}_{p}b^{u}_{p}R_{mqs,u}^{j}

, (3.36)

where we are summing over all indices, R^{j}_{mqs} represents the curvature
of ω_{X} and R^{j}_{mqs,u} its covariant derivative (with respect to ω_{X}). Since
these are ﬁxed tensors, we can use the Cauchy-Schwarz inequality and
(3.34), (3.35) to estimate the ﬁrst term on the right hand side of (3.36)

by 2Re

a^{i}_{kℓ}b^{m}_{k}b^{q}_{ℓ}b^{s}_{p}R^{j}_{mqs}a^{i}_{rp}a^{r}_{j}

≤C X

i,k,ℓ,r,p

|a^{i}_{kℓ}a^{i}_{rp}|
s λ_{r}

λ_{k}λ_{ℓ}λ_{p}

≤C

X

j

λ_{j}

1 2

X

q

1
λ_{q}

!^{3}_{2}
X

k,ℓ,r,p

X

i

|a^{i}_{kℓ}|^{2}

!^{1}_{2}
X

i

|a^{i}_{rp}|^{2}

!^{1}_{2}

=C(tr_{ω}_{X}ω˜_{t})^{1}^{2}(tr_{ω}_{˜}_{t}ω_{X})^{3}^{2}

X

i,k,ℓ

|a^{i}_{kℓ}|^{2}

1

2

X

i,r,p

|a^{i}_{rp}|^{2}

1 2

=CS(tr_{ω}_{X}ω˜_{t})^{1}^{2}(tr_{ω}_{˜}_{t}ω_{X})^{3}^{2}.

The second and third term in (3.36) are estimated similarly, while the fourth term can be bounded by

2Re

a^{i}_{kℓ}a^{i}_{j}b^{m}_{k}b^{q}_{ℓ}b^{s}_{p}b^{u}_{p}R^{j}_{mqs,u}

≤C X

i,k,ℓ,p

|a^{i}_{kℓ}|
s λ_{i}

λ_{k}λ_{ℓ}λ^{2}_{p}

≤C

X

j

λ_{j}

1 2

X

q

1
λ_{q}

!2

X

i,k,ℓ

|a^{i}_{kℓ}|

≤C√

S(tr_{ω}_{X}ω˜_{t})^{1}^{2}(tr_{ω}_{˜}_{t}ω_{X})^{2}.
Overall we can estimate

(3.37)

∆_{ω}_{˜}_{t}S ≥ −CS(tr_{ω}_{˜}_{t}ω_{X})^{3/2}(tr_{ω}_{X}ω˜_{t})^{1/2}−C√

S(tr_{ω}_{˜}_{t}ω_{X})^{2}(tr_{ω}_{X}ω˜_{t})^{1/2}.
On the other hand from [TWY, Lemma 3.3] we see that

∆_{ω}_{˜}_{t}tr_{ω}_{X}ω˜_{t}=a^{i}_{kℓ}a^{i}_{pℓ}a^{k}_{j}a^{p}_{j} +a^{i}_{j}a^{i}_{r}b^{q}_{ℓ}b^{s}_{ℓ}R^{r}_{jqs}

≥X

i,j,ℓ

|a^{i}_{jℓ}|^{2}λ_{j}−CX

i,ℓ

λ_{i}
λ_{ℓ}

≥ X

k

1
λ_{k}

!−1

X

i,j,ℓ

|a^{i}_{jℓ}|^{2}−C X

p

λ_{p}

! X

q

1 λq

!

= S

tr_{ω}_{˜}_{t}ω_{X} −C(tr_{ω}_{˜}_{t}ω_{X})(tr_{ω}_{X}ω˜_{t}).

(3.38)

We now insert (3.29), (3.30) in (3.37), (3.38) and get
(3.39) ∆_{ω}_{˜}_{t}S ≥ −CF(2C_{0})

t^{3/2} S − CF(5C_{0}/2)
t^{2}

√S,

∆_{ω}_{˜}_{t}tr_{ω}_{X}ω˜_{t}≥ tF(−C_{0})

C S − CF(2C_{0})
t .