Geometric estimates for complex Monge–Ampère equations

Geometric estimates for complex Monge–Ampère equations

ByXin Fuat Piscataway,Bin Guoat New York andJian Songat Piscataway

Abstract. We prove uniform gradient and diameter estimates for a family of geometric complex Monge–Ampère equations. Such estimates can be applied to study geometric regu- larity of singular solutions of complex Monge–Ampère equations. We also prove a uniform diameter estimate for collapsing families of twisted Kähler–Einstein metrics on Kähler mani- folds of nonnegative Kodaira dimensions.

1. Introduction

Complex Monge–Ampère equations are a fundamental tool to study Kähler geometry and, in particular, canonical Kähler metrics of Einstein type on smooth and singular Kähler varieties. Yau’s solution to the Calabi conjecture establishes the existence of Ricci flat Kähler metrics on Kähler manifolds of vanishing first Chern class by a priori estimates for complex Monge–Ampère equations [40].

Let.X; /be a Kähler manifold of complex dimension nequipped with a Kähler met- ric. We consider the following complex Monge–Ampère equation:

(1.1) .Ci𝜕𝜕'/N ⁿDe ^fn;

wheref 2C¹.X /satisfies the normalization condition Z

X

e ^fnD Z

X

ⁿDŒ ⁿ:

In the deep work of Kolodziej [15], Yau’sC⁰-estimate for solutions of equation (1.1) is tremen- dously improved by applying the pluripotential theory and it has important applications for sin- gular and degenerate geometric complex Monge–Ampère equations. More precisely, suppose the right-hand side of equation (1.1) satisfies the followingL^p bound:

Z

X

e ^pfnK for somep > 1I

Research supported in part by National Science Foundation grants DMS-1406124 and DMS-1710500.

then there existsC DC.X; ; p; K/ > 0such that any solution'of equation (1.1) satisfies the followingL¹-estimate:

k' sup

X

'kL¹.X /C:

In particular, equation (1.1) admits a unique continuous solution in PSH.X; / as long as e ^f 2L^p.X; ⁿ/without any additional regularity assumption forf. In [7, 16], it is shown that the bounded solution is also Hölder continuous and the Hölder exponent only depends only onnandp. However, in general the solution is not uniformly Lipschitz continuous (see e.g. [7]).

Complex Monge–Ampère equations are closely related to geometric equations of Einstein type, and in many geometric settings, one makes assumption on a uniform lower bound of the Ricci curvature. Therefore it is natural to consider the family of volume measures, whose curvature is uniformly bounded below. More precisely, we letDe ^fnbe a smooth volume form onX such that

(1.2) Ric./D i𝜕𝜕Nlog A

for some fixed constantA0. This is equivalent to saying, i𝜕N𝜕f Ric. / A;

or

f 2PSH.X;Ric. /CA /:

We will explain one of the motivations for condition (1.2) by some examples. Let¹Eiº^IiD1

and¹Fjºj^JD1be two families of effective divisors ofX. LetE_iandF_j be the defining sections forEi andFj, respectively, andhEi andhFj smooth hermitian metrics for the line bundles associated toEiandFj, respectively. In [40], Yau considers the following degenerate complex Monge–Ampère equations:

(1.3) .Ci𝜕𝜕'/N ⁿD

P^I

iD1j_Eij^2ˇ_hEiⁱ PJ

jD1jF_jj^2˛_hFj^j

! ⁿ;

where˛j; ˇi > 0, and various estimates are derived [40] assuming certain bounds on the degen- erate right-hand side of equation (1.3).

If we consider the following case:

(1.4) .Ci𝜕𝜕'/N ⁿD ⁿ

PJ

jD1jFjj^2˛_h ^j

Fj

;

the volume measure will blow up along common zeros of¹Fjºj^JD1. If the volume measure on the right-hand side of equation (1.4) isL^p-integrable for somep > 1, i.e.,

D

J

X

jD1

jF_jj^2˛_hFj^j

! 1

ⁿ

satisfies

 ⁿ D

J

X

jD1

j_Fjj^2˛_hFj^j

! 1

2L^p.X; ⁿ/ for somep > 1;

Z

X

D Z

X

ⁿ;

then there exists a unique (up to a constant translation) continuous solution of (1.4). Further- more,can be approximated by smooth volume formsj (cf. [6]) satisfying

Ric.j/ .ACA⁰/;

j

ⁿ

_Lp.X;ⁿ/

 ⁿ

_Lp.X;ⁿ/

; Z

X

j D Z

X

ⁿ

for some fixedA⁰0. Therefore condition (1.2) is a natural generalization of the above case.

In the special case when¹Fjºj^JD1is a union of smooth divisors with simple normal crossings and each ˛j 2.0; 1/, the solution of equation (1.4) has conical singularities of cone angle of2.1 ˛j/alongFj,j D1; : : : ; J.

We now state the first result of the paper.

Theorem 1.1. Let.X; /be an Kähler manifold of complex dimensionnequipped with a Kähler metric. We consider the following complex Monge–Ampère equation:

(1.5) . Ci𝜕𝜕'/N ⁿDe^';

whereD0or1, andis a smooth volume form satisfyingR

XDR

Xⁿ. If (1.6)

Z

X

 ⁿ

p

ⁿK; Ric./D i𝜕𝜕Nlog A;

for some p > 1, K > 0 and A0, then there exists a constant C DC.X; ; p; K; A/ > 0 such that the solution ' of equation(1.5) and the Kähler metric g associated to the Kähler form!DCi𝜕𝜕'N satisfy the following estimates:

(1) k' sup_X'kL¹.X /C krg'kL¹.X;g/C, (2) Ric.g/ C g,

(3) Diam.X; g/C.

If we write De ^fn, assumption (1.6) in Theorem 1.1 on  is equivalent to the following onf:

e ^f 2L^p.X; /;

Z

X

e ^f DŒ ⁿ; f 2PSH.X;Ric. /CA /:

The functionf is uniformly bounded above by the plurisubharmonicity and the Kähler met- ricgassociated to!DCi𝜕𝜕'N is bounded below by a fixed multiple of (see Lemma 2.2).

However, one cannot expect thatg is bounded from above sincef is not uniformly bounded above as in the example of equation (1.4). Fortunately, we can bound the diameter of.X; g/

uniformly by Theorem 1.1.

The gradient estimate in Theorem 1.1 is a generalization of the gradient estimate in [26].

The new insight in our approach is that one should estimate gradient and higher-order estimates of the potential functions with respect to the new metric instead of a fixed reference metric for geometric complex Monge–Ampère equations such as those studied in Theorem 1.1. We refer interested readers to [3, 22, 23] for gradient estimates for complex Monge–Ampère equations with respect to various background metrics.

Let M.X; ; p; K; A/ be the space of all solutions of equation (1.5), where  satis- fies assumption (1.6) in Theorem 1.1. We also identifyM.X; ; p; K; A/ with the space of

Kähler forms!DCi𝜕N𝜕'for' 2M.X; ; p; K; A/. An immediate consequence of Theo- rem 1.1 is a uniform noncollapsing condition for M.X; ; p; K; A/. More precisely, there exists a constant C DC.X; ; p; K; A/ > 0 such that for all Kähler metric g associated to

!2M.X; ; p; K; A/and for any pointx2X,0 < r < 1, (1.7) C ¹r²ⁿVolg.Bg.x; r//C r²ⁿ; whereBg.x; r/is the geodesic ball centered atxwith radiusrin.X; g/.

Combining the lower bound of Ricci curvature and the noncollapsing condition (1.7), we can apply the theory of degeneration of Riemannian manifolds [5] so that any sequence of Kähler manifolds.X; gj/2M.X; ; p; K; A/, after passing to a subsequence, converges to a compact metric space.X₁; d₁/with well-defined tangent cones of Hausdorff dimension2n at each point inX₁. In the case of equation (1.4), we believe the solution induces a unique Riemannian metric space homeomorphic to the original manifoldX and all tangent cones are unique. If this is true, one might even be able to establish higher-order expansions for the solution. The ultimate goal of our approach is to construct canonical domains and equations on the blow-up of solutions for geometric degenerate complex Monge–Ampère equations, by degeneration of Riemannian manifolds.

We also remark that if we replace the lower bound for Ric./by an upper bound Ric./A

in assumption (1.6) of Theorem 1.1, we can still obtain a uniform diameter upper bound. This in fact easily follows from the argument for the second-order estimates of Yau [40] and Aubin [1].

We will also use similar techniques in the proof of Theorem 1.1 to obtain diameter esti- mates in more geometric settings. Before that, let us introduce a few necessary and well-known notions in complex geometry.

Definition 1.1. LetX be a Kähler manifold and ˛2H².X;R/\H^1;1.X;R/. Then the class˛is nef if˛CAis a Kähler class for any Kähler classA.

Definition 1.2. LetX be a Kähler manifold of complex dimensionnand let the class

˛2H².X;R/\H^1;1.X;R/be nef. The numerical dimension of the class˛is given by .˛/Dmax¹kD0; 1; : : : ; nW˛^k ¤0inH^2k.X;R/ºI

when.˛/Dn, the class˛is said to be big.

The numerical dimension.˛/is always no greater than dimC.X /.

When the canonical bundleKX is nef,X is said to be a minimal model. The abundance conjecture in birational geometry predicts that the canonical line bundle is always semi-ample (i.e., a sufficiently large power of the canonical line bundle is globally generated) if it is nef.

Definition 1.3. Let # be a smooth real-valued closed .1; 1/-form on a Kähler mani- foldX. The extremal functionV associated to the form#is defined by

V .z/Dsup

°

.z/W#Ci𝜕𝜕N 0; sup

X

D0±

for allz2X.

Any 2PSH.X; #/is said to have minimal singularities defined by Demailly (cf. [2]) if V is bounded.

Let.X; /be a Kähler manifold of complex dimension nequipped with a Kähler met- ric . Suppose is a real-valued smooth closed .1; 1/-form and its class Œ is nef and of numerical dimension. We consider the following family of complex Monge–Ampère equa- tions:

(1.8) .Ct Ci𝜕𝜕'N t/ⁿDtⁿ e^'tCct fort 2.0; 1;

whereD0, or1, andct is a normalizing constant such that (1.9)

Z

X

tⁿ e^ctD Z

X

.Ct /ⁿ:

Straightforward calculations show thatct is uniformly bounded for t 2.0; 1. The following proposition generalizes the result in [4, 10, 15, 42] by studying a family of collapsing complex Monge–Ampère equations. It also generalizes the results in [8, 9, 17] for the case when the limiting reference form is semi-positive.

Proposition 1.1. We consider equation (1.8) with the normalization condition (1.9).

Suppose the volume measuresatisfies Z

X

 ⁿ

p

ⁿK

for some p > 1and K > 0. Then there exists a unique 't 2PSH.X; Ct / up to a con- stant translation solving equation(1.8)for allt 2.0; 1. Furthermore, there exists a constant C DC.X; ; ; p; K/ > 0such that for allt 2.0; 1,

't sup

X

't

Vt

_{L1.X /} C;

whereVt is the extremal function associated toCt as in Definition1.3.

Proposition 1.1 can be applied to generalize Theorem 1.1, especially for minimal Kähler manifolds in a geometric setting.

Theorem 1.2. SupposeX is a smooth minimal model equipped with a smooth Kähler form. For anyt > 0, there exists a unique smooth twisted Kähler–Einstein metricgt onX satisfying

(1.10) Ric.gt/D gt Ct :

There exists a constantC DC.X; / > 0such that for allt 2.0; 1, Diam.X; gt/C:

Furthermore, for any tj !0, after passing to a subsequence, the twisted Kähler–Einstein manifolds.X; gtj/converge in Gromov–Hausdorff topology to a compact metric length space .Z; d_Z/. The Kähler forms !tj associated to gtj converge in distribution to a nonnegative closed current

e!DCi𝜕𝜕Ne'

for somee'2PSH.X; / of minimal singularities, where 2ŒKXis a fixed smooth closed .1; 1/-form.

Both Theorem 1.1 and Theorem 1.2 are generalization and improvement for the tech- niques developed in [26] for diameter and distance estimates. With the additional bounds on the volume measure, we transform Kolodziej’s analyticL¹-estimate to a geometric diameter estimate. The relation between analytic estimates of Kähler potentials and geometric estimates for distance functions was also studied in [20]. It is a natural question to ask how the metric space.Z; d_Z/is related to the currente! onX. We conjecturee!is smooth on an open dense set ofX and its metric completion coincides with.Z; d_Z/. However, at this moment, we do not even know the Hausdorff dimension or uniqueness of.Z; d_Z/.

When X is a minimal model of general type, Theorem 1.2 is proved in [26, 27] and the result in [36] shows that the singular set is closed and of Hausdorff dimension no greater than2n 4.

We can also replace the smooth Kähler formin Theorem 1.2 by Dirac measures along effective divisors. For example, if¹Ejºj^JD1is a union of smooth divisors with normal crossings

and J

X

jD1

ajEj

is an ampleQ-divisor with someaj 2.0; 1/forj D1; : : : ; J, then Theorem 1.2 also holds if we let DPJ

jD1ajŒEj:In this case, the metricgt is a conical Kähler–Einstein metric with cone angles of2.1 aj/along each complex hypersurfaceEj.

A special case of the abundance conjecture is proved by Kawamata [14] for minimal models of general type. WhenX is a smooth minimal model of general type, it is recently proved by the third named author [27] that the limiting metric space.Z; d_Z/in Theorem 1.2 is unique and is homeomorphic to the algebraic canonical modelX_canofX. This gives an analytic proof of Kawamata’s result using complex Monge–Ampère equations, Riemannian geometry and geometricL²-estimates. Theorem 1.2 also provides a Riemannian geometric model for the non-general type case. This analytic approach will shed light on the abundance conjecture if such a metric model is unique with reasonably good understanding of its tangle cones.

Theorem 1.2 can also be easily generalized to a Calabi–Yau manifoldX equipped with a nef line bundleLoverX of.L/D.

Our final result assumes semi-ampleness for the canonical line bundle and aims to con- nect the algebraic canonical models to geometric canonical models. LetXbe a Kähler manifold of complex dimensionn. If the canonical bundleKX is semi-ample, the pluricanonical system induces a holomorphic surjective map

ˆWX !X_can

from X to its unique canonical model X_can. In particular, dimCX_canD.X /. Let S be the set which consists of all singular fibers of ˆ together with ˆ ¹.S_Xcan/, where S_Xcan is the singular set of X_can. The general fiber of ˆ is a smooth Calabi–Yau manifold of complex dimensionn .X /. It is proved in [26, 27] that there exists a unique twisted Kähler–Einstein current!_canonX_cansatisfying

Ric.!_can/D !_canC!_WP;

where ˆ!_can2 c1.X / and !_WP is the Weil–Petersson metric for the variation of the Calabi–Yau fibers. In particular,!_canhas bounded local potentials and is smooth onX_cannS_can. We letg_canbe the smooth Kähler metric associated to!_canonX_cannS_can.

Theorem 1.3. SupposeX is a projective manifold of complex dimensionn equipped with a Kähler metric. If the canonical bundleKX is semi-ample and.KX/D2N, then for the twisted Kähler–Einstein metricsgt satisfying

Ric.gt/D gt Ct ; t 2.0; 1;

the following hold:

(1) There existsC > 0such that for allt 2.0; 1, Diam.X; gt/C:

(2) Let!t be the Kähler form associated togt. For any compact subsetK XnS, we have

kgt ˆg_cankC⁰.K; / !0 ast !0:

(3) The rescaled metricst ¹!tj^Xy converge uniformly to a Ricci-flat Kähler metric!C Y;y

on the fiberXy Dˆ ¹.y/for anyy 2X_cannˆ.S/, ast !0.

(4) For any sequence tj !0, after passing to a subsequence, the manifolds .X; gt_j/ con- verge in Gromov–Hausdorff topology to a compact metric space.Z; d_Z/. Furthermore, X_cannS_can is embedded as an open subset in the regular partR2 of.Z; d_Z/ and the manifold.X_cannS_can; !_can/is locally isometric to its open image.

In particular, if D1, then.Z; d_Z/is homeomorphic toX_can, with the regular part being open and dense, and each tangent cone being a metric cone onCwith cone angle less than or equal to2.

We remark that a special case of Theorem 1.3 is proved in [41] with a different approach for dim_CX D2. In general, the collapsing theory in Riemannian geometry has not been fully developed except in lower dimensions. In the Kähler case, one hopes the rigidity properties can help us understand the collapsing behavior for Kähler metrics of Einstein type as well as long time solutions of the Kähler–Ricci flow on algebraic minimal models. Key analytic and geometric estimates in the proof of (2) in Theorem 1.3 are established in [29, 30] for the collapsing long time solutions of the Kähler–Ricci flow and its elliptic analogues. The proof for (3) and (4) is a technical modification of various local results of [11, 12, 37, 38], where collapsing behavior for families of Ricci-flat Calabi–Yau manifolds is comprehensively studied. Theorem 1.3 should also hold for Kähler manifolds with some additional arguments.

Finally, we will also apply our method to a continuity scheme proposed in [18] to study singularities arising from contraction of projective manifolds. This is an alternative approach for the analytic minimal model program developed in [29–31] to understand birational trans- formations via analytic and geometric methods [25, 28, 32–34]. Compared to the Kähler–Ricci flow, such a scheme has the advantage of prescribed Ricci lower bounds and so one can apply the Cheeger–Colding theory for degeneration of Riemannian manifolds, on the other hand, it loses the canonical soliton structure for the analytic transition of singularities corresponding to birational surgeries such as flips.

Let X be a projective manifold of complex dimension n. We choose an ample line bundle L on X and we can assume that L KX is ample, otherwise we can replace L by a sufficiently large power of L. We choose 2ŒL KXto be a smooth Kähler form and

consider the following curvature equation:

(1.11) Ric.gt/D gtCt ; t 2Œ0; 1:

Let

t_minDinf¹t 2Œ0; 1Wequation (1.11) is solvable attº:

It is straightforward to verify thatt_min< 1by the usual continuity method (cf. [18]). The goal is to solve equation (1.11) for allt 2.0; 1, however, one might have to stop att Dt_minwhen KX is not nef.

Theorem 1.4. Letgtthe solution of equation(1.11)fort 2.t_min; 1. There exists a con- stantC DC.X; / > 0such that for anyt 2.t_min; 1,

Diam.X; gt/C:

Theorem 1.2 is a special case of Theorem 1.4 whent_minD0(cf. [26]). Whent_min> 0, Theorem 1.4 is also proved in [19] with the additional assumption thatt_minLC.1 t_min/KX

is semi-ample and big. The diameter estimate immediately allows one to identify the geomet- ric limit as a compact metric length space when t !t_min. In particular, it is shown in [19]

that the limiting metric space is homeomorphic to the projective variety from the contraction induced by theQ-line bundlet_minLC.1 t_min/KX when it is big and semi-ample. One can also use Theorem 1.4 to obtain a weaker version of Kawamata’s base point free theorem in the minimal model theory (cf. [13]). If t_minLC.1 t_min/KX is not big, our diameter esti- mate still holds and we conjecture the limiting collapsed metric space of.X; gt/ast !t_min is unique and is homeomorphic a lower dimensional projective variety from the contraction induced byt_minLC.1 t_min/KX.

2. Proof of Theorem 1.1

Throughout this section, we let' 2PSH.X; /be the solution of equation (1.5) satis- fying condition (1.6) in Theorem 1.1. We let !DCi𝜕𝜕N' and letg be the Kähler metric associated to!.

Lemma 2.1. There exists a constantC DC.X; ; p; K/ > 0such that k' sup

X

'kL¹.X /C:

Proof. TheL¹-estimate immediately follows from Kolodziej’s theorem [15].

The following is a result similar to Schwarz lemma [39].

Lemma 2.2. There exists a constantC DC.X; ; p; K; A/ > 0such that

! C :

Proof. There exists a constantC DC.X; ; A/ > 0such that

!log tr!. / C C tr!. /;

where!is the Laplace operator associated with!. Then let H Dlog tr!. / B.' sup

X

'/

for someB > 2C. Then

!H C tr!. / C:

It follows from maximum principle and theL¹-estimate in Lemma 2.1 that sup

X

tr! C :

Lemma 2.2 immediately gives the uniform Ricci lower bound.

Lemma 2.3. There exists a constantC D.X; ; p; K; A/ > 0such that Ric.g/ C g:

Proof. We calculate

Ric.g/D gCRic./C g .A / C g

for some fixed constantC > 0by Lemma 2.2.

We will now prove the uniform diameter bound.

Lemma 2.4. There exists a constantC D.X; ; p; K; A/ > 0such that Diam.X; g/C:

Proof. We first fix a sufficiently smallD.p/ > 0so thatp > 1. Without loss of generality we may assume Diam.X; g/DDfor someD100. Let WŒ0; D!X be a nor- mal minimal geodesic with respect to the metricg and choose the points ¹xi D.6i /º^ŒD=6_iD0 . The balls¹Bg.xi; 3/º^ŒD=6iD0 are disjoint, so

ŒD=6

X

iD0

Volⁿ.Bg.xi; 3//CVol.Bg.xi; 3//

Z

X

ⁿCD2V;

hence there exists a geodesic ballBg.xi; 3/such that

Volⁿ.Bg.xi; 3//CVol.Bg.xi; 3//12VD ¹: We fix such anxi and construct a cut-off function.x/D.r.x//0with

r.x/Ddg.x; xi/ such that

D1 onBg.xi; 1/; D0 outsideBg.xi; 2/;

and

2Œ0; 1; ¹.⁰/² 10; j⁰⁰j 10:

Define a piecewise linear continuous function FQ WR!R such that F .t /DD^{p.p /} when t 2Œ0; 2, F .t /Q a when t 3 and F .t /Q is linear when t 2Œ2; 3, where a > 0 is a con- stant to be determined. DenoteF .x/D QF .r.x//; thenF aoutsideBg.xi; 3/,F DD^{p.p /} onBg.xi; 2/. We choose the constanta > 0so thatR

XF DŒ ⁿDV. We observe that V DaVol.X nBg.xi; 3//C

Z

Bg.x_i;3/

F V .1 12D ¹/a V 2a so0 < a 2. Then it follows that

Z

X

F  ⁿ

p

ⁿ Z

X

F^{p.p /n} _p Z

X

 ⁿ

p

^{n p}_p

C

for someC DC.X; ; p; K/ > 0.

We now consider the equation

. Ci𝜕𝜕/N ⁿDeF :

By similar arguments as before,k sup_XkL¹ C DC.X; ; p; K/. LetgO D Ci𝜕𝜕N. Then onBg.xi; 2/,

Ric.g/O D gOCRic./C; Ric.g/D gCRic./C: In particular,

glog!Oⁿ

!ⁿ D nCtrg.g/;O whereg D!. Let

H D

log!Oⁿ

!ⁿ

' sup

X

'

sup

X

:

OnBg.xi; 2/;we have

gH D .C1/nC.C1/trg.g/O 2nCn !Oⁿ

!ⁿ _n¹

:

In general, on the support of, we have

gH

2nCn !Oⁿ

!^{n 1}_n

C2 ¹Re.rH r/ 2Hjrj²

² C ¹Hg ¹

C ²e^nH C2Re.rH r/ 2Hjrj²

CHg 2n²

:

We may assume sup_XH > 0, otherwise we already have upper bound ofH. The maximum ofH must lie atBg.xi; 2/and at this point

gH 0; jrHj² D0:

By Laplacian comparison we have

gD⁰rC⁰⁰ C; jrj²

D .⁰/²

C:

So at the maximum ofH, it holds that

0C ²e^nH CH 2nCH² CH 2n;

therefore we have sup_XH C. In particular, on the ballBg.xi; 1/where 1, it follows that^!_!^Onn C. From the definition of!O and!,

C !Oⁿ

!ⁿ DD^{p.p /}e^{. '/}: Combined with theL¹-estimate ofand', we conclude that

DC DC.n; p; ; A; K/:

Lemma 2.5. There exists a constantC D.X; ; p; K; A/ > 0such that sup

X jr^g'j^g C:

Proof. Straightforward calculations show that

g' Dn trg. /;

gjr'j²g D jrr'j²C jr Nr'j²Cg^ilNg^kjNR_ijN'_k'_lN 2r' rtrg. / jrr'j²C jr Nr'j² Cjr'j² 2r' rtrg. /;

and

gtrg Dtrg glog trgCjrtrgj²

trg C Cc0jrtrgj² for some uniform constantc0; C > 0. We choose constants˛andBsatisfying

˛ > 4c₀¹> 4; B >sup

X

'C1

and define

H D jr'j²

B ' C˛trg: Then we have

H jrr'j²C jr Nr'j²

B ' C jr'j²

B '

jr'j².trg n/

.B '/² (2.1)

2.1C˛/hr';rtrgi

B ' ˛C C˛c0jrtrgj²C2 r'

B ';rH

:

We may assume at the maximum point z_max of H,jr'j> ˛ and H > 0, otherwise we are done. Atz_max,

rH D0; H 0 and so atz_max,

rjr'j D 1 2

H r'

jr'j ˛.B '/rtrg

jr'j C˛trgr' jr'j

:

By Kato’s inequality

jrjr'jj² jrr'j²C jr Nr'j²

2 ;

it follows that

jrr'j²C jr Nr'j²

B ' 1

2.B '/

H²C˛².B '/².trg /²C˛²jrtrgj² jr'j² (2.2)

2˛H.B '/trg 2˛Hjrtrgj jr'j 2˛².B '/trgjrtrgj

jr'j

H²

4.B '/ CH jrtrgj²

B ' Cjrtrgj

for some uniform constant C > 0. After substituting inequality (2.2) to (2.1) and applying Cauchy–Schwarz inequality, we have atz_max

0 H²

4.B '/ CH C 2jrtrgj²

B ' Cjrtrgj C4jrtrgj² H²

4.B '/ CH C

for some uniform constantC > 0. Therefore maxXHC for someC DC.X; ; ; A; p; K/.

The lemma then immediately follows from Lemma 2.1 and Lemma 2.2.

3. Proof of Proposition 1.1

In this section, we will prove Proposition 1.1 by applying the techniques in [4, 8, 10, 15].

LetX be a Kähler manifold of dimensionn. Suppose˛ is nef class onX of numerical dimension 0. Let 2˛ be a smooth closed .1; 1/-form. We define the extremal func- tionVby

VDsup¹ WCi𝜕𝜕N 0; 0º:

Letbe a fixed smooth Kähler metric onX. Then we define the perturbed extremal functionVt

fort 2.0; 1by

Vt Dsup¹ WCt Ci𝜕𝜕N 0; 0º: The above extremal functions were introduced in [4] when˛is big.

We first rewrite equation (1.8) forD0as follows:

(3.1) .Ct Ci𝜕𝜕'N t/ⁿDtⁿ e ^fCctn; sup

X

't D0; t 2.0; 1;

by lettingDe ^fn, wherect is the normalizing constant satisfying tⁿ

Z

X

e ^fCctnD Z

X

.Ct /ⁿ:

The functionf satisfies the following uniform bound:

Z

X

e ^pfnK

for somep > 1andK > 0.

The following definition is an extension of the capacity introduced in [4, 8, 10, 15].

Definition 3.1. We define the capacity Cap_t.K/for a subsetK X by Cap_t.K/Dsup

²Z

K

.t Ci𝜕𝜕u/N ⁿWu2PSH.X; t/; 0u Vt 1

³

;

wheret DCt is the reference metric in (3.1). We also define the extremal functionV_t;K by

Vt;K Dsup¹u2PSH.X; t/Wu0onKº: IfKis open, then we have

(1) V_t;K 2PSH.X; t/\L¹.X /, (2) .tCi𝜕𝜕VN t;K/ⁿD0onXnK.

Lemma 3.1. Let 't be the solution to (3.1). Then there exist ıDı.X; ; / > 0 and C DC.X; ; ; p; K/ > 0such that for any open setK X andt 2.0; 1,

1 Œⁿ_t

Z

K

.t Ci𝜕𝜕N't/ⁿC e ^ı

Œnt  Capt .K/

n¹

:

Proof. SinceŒ^mD0forC1mn, it follows that Œⁿ_tD

Z

X

ⁿ_t D Z

X n

X

kD0

n k

!

^k^t^{n kn k}

D Z

X

X

kD0

n k

!

^k^t^{n kn k} DO.tⁿ /:

It follows that the normalizing constantctin (3.1) is uniform bounded. LetM_t;K Dsup_XV_t;K. Then we have

1 Œⁿ_t

Z

K

.t Ci𝜕𝜕'N t/ⁿD tⁿ e^ct Œⁿ_t

Z

K

e ^fn

tⁿ e^ct Œⁿ_t

Z

K

e ^fe ^{ıVt;Kq n} (sinceV_t;K 0onK) tⁿ e^ct

Œⁿ_t e ^ıMt;Kq Z

X

e ^fe ^{ı.Vt;K Mt;K}

/

q ⁿ

tⁿ e^ct

Œⁿ_t e ^ıMt;Kq Z

X

e ^pfn _p¹Z

X

e ^{ı.Vt;K Mt;K/n 1}_q

C e ^ıMt;Kq ;

where_p¹ C¹_q D1. Obviously, there exists D.X; ; / > 0such that for allt 2.0; 1, V_t;K 2PSH.X; /:

We apply the global Hörmander’s estimate ([35]) so that there existsıDı.X; ; / > 0such

that Z

X

e ^{ı.Vt;K supXVt;K/n}C_ı:

To complete the proof, it suffices to show

(3.2) Mt;K C1

Œⁿ_t Cap_t.K/

¹_n :

First we observe that by definition sup

X

.V_t;K sup

X

V_t;K/ Vt

0;

since Vt;K sup_XVt;K 2PSH.X; t/ is nonpositive. On the other hand, Vt;K Vt. This immediately implies that

(3.3) 0Vt;K Vt sup

X

Vt;K DMt;K:

We break the rest of the proof into two cases.

The case whenM_t;K > 1. We let

t;K DM_t;K¹.V_t;K Vt/CVt: Then

Vt t;K VtC1 and by (3.3),

1

M_t;Kⁿ D 1 M_t;Kⁿ

R

X.t Ci𝜕𝜕VN t;K/ⁿ Œⁿ_t

(3.4)

D 1 Œⁿ_t

Z

K

M_t;K¹tCi𝜕N𝜕.M_t;K¹Vt;K/n

1 Œⁿ_t

Z

K

M_t;K¹t Ci𝜕𝜕.MN _t;K¹Vt;K/C.1 M_t;K¹/.t Ci𝜕𝜕VN t//n

D 1 Œⁿ_t

Z

K

.t Ci𝜕𝜕 N t;K/ⁿ

Cap_t.K/ Œⁿ_t :

The case whenM_t;K 1. By (3.3), 0V_t;K Vt sup

X

V_t;K DM_t;K 1:

Now

(3.5) Œⁿ_tD

Z

K

.t Ci𝜕𝜕VN t;K/ⁿCap_t.K/:

So in this case

Œⁿ_t

Cap_t.K/ 1:

Combining (3.4) and (3.5), (3.2) holds and we complete the proof of Lemma 3.1.

The following is an immediate corollary of Lemma 3.1.

Corollary 3.1. There existsC DC.X; ; ; p; K/ > 0such that for allt 2.0; 1, we have

1 Œⁿ_t

Z

K

.t Ci𝜕𝜕'N t/ⁿC

Cap_t.K/ Œⁿ_t

2

:

Proof. This follows from Lemma 3.1 and the elementary inequality that x²e ^ıx

1n

C

for some uniformC > 0and allx2.0;1/.

Lemma 3.2. Letu2PSH.X; t/\L¹.X /. For anys > 0,0r 1andt 2.0; 1, we have

(3.6) rⁿCap_t.u Vt < s r/

Z

¹u Vt< sº

.t Ci𝜕𝜕u/N ⁿ:

Proof. For any 2PSH.X; t/with0 Vt 1, we have rⁿ

Z

¹u Vt< s rº

.tCi𝜕𝜕N/ⁿ

D Z

¹u Vt< s rº

.rt Ci𝜕𝜕.r//N ⁿ

Z

¹u Vt< s rº

.t Ci𝜕𝜕.r/N Ci𝜕𝜕.1N r/Vt/ⁿ

Z

¹u Vt< s rCr. Vt/º

.t Ci𝜕𝜕.rN C.1 r/Vt s r//ⁿ

Z

¹u<rC.1 r/V_t s rº

.tCi𝜕𝜕u/N ⁿ

Z

¹u<Vt sº

.t Ci𝜕𝜕u/N ⁿ:

The third inequality follows from the comparison principle and the last inequality follows from the fact that

rC.1 r/Vt s r Dr. Vt 1/CVt s < Vt s:

Taking supremum of all 2PSH.X; t/with0 Vt 1, we get (3.6).

Lemma 3.3. Let 't be the solution to equation (3.1). Then there exists a constant C DC.X; ; ; p; K/ > 0such that for alls > 1,

1

Œⁿ_tCap_t.¹'t Vt < sº/ C .s 1/^1q

;

where _p¹ C¹_q D1.

Proof. Applying Lemma 3.2 touD't andrD1, we have 1

Œⁿ_tCap_t.¹'t Vt < sº/ 1 Œⁿ_t

Z

¹'t Vt< .s 1/º

.t Ci𝜕𝜕'N t/ⁿ

D 1 Œⁿ_t

Z

¹'t Vt< .s 1/º

tⁿ e ^fCctn

C

.s 1/^1q Z

¹'t Vt< .s 1/º

. 'tCVt/^1qe ^fn

C

.s 1/^1q Z

¹'_t V_t< .s 1/º

e ^pfn _p¹

Z

¹'t Vt< .s 1/º

. 't CVt/ⁿ _q¹

C

.s 1/^1q Z

X

. 't/^{n 1}_q

;

where in the last inequality we use the assumption thate ^f 2L^p.ⁿ/,Vt 0and't 0. On the other hand, since't 2PSH.X; t/PSH.X; C /for some largeC > 0and sup_X't D0, it follows from Green’s formula that

Z

X

. 't/ⁿC

for some uniform constantC. The lemma follows by combining the inequalities above.

The following lemma is well known and its proof can be found, e.g., in [10, 15].

Lemma 3.4. Let F WŒ0;1/!Œ0;1/ be a non-increasing right-continuous function satisfyinglims!1F .s/D0. If there exist˛; A > 0such that for alls > 0and0r 1,

rF .sCr/A.F .s//^1C˛; then there existsS DS.s0; ˛; A/such that

F .s/D0

for allsS, wheres0is the smallestssatisfying.F .s//^˛ .2A/ ¹. Proof of Proposition1.1. Define for each fixedt 2.0; 1,

F .s/D

Cap_t.¹'t Vt < sº/ Œtⁿ

¹_n :

By Corollary 3.1 and Lemma 3.2 applied to the function't, we have rF .sCr/AF .s/² for allr2Œ0; 1; s > 0;

for some uniform constantA > 0independent oft 2.0; 1.

Lemma 3.3 implies that lims!1F .s/D0and thes0in Lemma 3.4 can be taken as less than.2AC /^q, which is a uniform constant. It follows from Lemma 3.4 thatF .s/D0 for all s > S, whereS 2Cs0. On the other hand, if Cap_t.¹'t Vt < sº/D0, by Lemma 3.1 and equation (3.1), we have

Z

¹'t Vt< sº

e ^fnD0;

hence the set¹'t Vt < sº D ;. Thus infX.'t Vt/ S. Thus we finish the proof of Proposition 1.1.

Therefore we have proved Proposition 1.1 whenD0. We finish this section by prov- ing the case whenD1. To this end, we consider the following complex Monge–Ampère equations fort2.0; 1:

.Ct Ci𝜕𝜕'N t/ⁿDtⁿ e^{'t fCctn}; wheref 2C¹.X /andct is the normalizing constant satisfying

tⁿ Z

X

e ^fCctnD Z

X

.Ct /ⁿ:

Corollary 3.2. Ifke ^fkL^p.X;ⁿ/K, forp > 1andK > 0, Then there exists a con- stantC DC.X; ; ; p; K/ > 0such that

k't Vtk^L1 C:

Proof. Since for eacht > 0, it is proved in [2] thatVt isC^1;˛.X; /, we can always find Wt 2C¹.X /such that sup_XjVt Wtj 1:Furthermore,Vt is uniformly bounded above for allt 2.0; 1. We let t be the solution of

.tCi𝜕N𝜕 t/ⁿDtⁿ e ^fCctCWtn; sup

X

t D0:

and

ut D't t: Then

.tCi𝜕𝜕N tCi𝜕𝜕Nut/ⁿ

.tCi𝜕𝜕 N t/ⁿ De^{utC t Wt}:

Since sup_Xj ^t Wtj sup_Xj ^t Vtj C1, the maximum principle immediately implies that kutkL¹.X / k ^t VtkL¹.X /C1

and so

k't VtkL¹.X / 2k t VtkL¹.X /C1:

4. Proof of Theorem 1.2

LetX be a Kähler manifold;X is said to be a minimal model if the canonical bundleK_X is nef. The numerical dimension ofKX is given by

.KX/Dmax¹mD0; : : : ; nWŒKX^m¤0inH^m;m.X;C/º:

Letbe a smooth Kähler form on a minimal modelXof complex dimensionn. Let D.X /, the numerical dimension ofKX. Letbe a smooth volume form onX. We letbe defined by

Di𝜕𝜕Nlog2K_X:

We consider the following Monge–Ampère equation fort 2.0;1/, (4.1) .Ct Ci𝜕𝜕'N t/ⁿDtⁿ e^'t:

SinceKX is nef,ŒCt is a Kähler class for anyt > 0. By Aubin and Yau’s theorem, there exists a unique smooth solution't solving (4.1) for allt > 0. Let!t DCt Ci𝜕𝜕'. ThenN

!t satisfies

Ric.!t/D !t Ct :

In particular, any Kähler metric satisfying the above twisted Kähler–Einstein equation must coincide with!t.

Lemma 4.1. There exists a constantC > 0such that for allt 2.0; 1, C ¹tⁿ ŒCt ⁿC tⁿ :

Proof. First we note thatŒŒ ⁿ > 0becauseŒ¤0andŒis nef. Then ŒCt ⁿDtⁿ n

!

ŒŒ ⁿ Ct^{n C1}

n

X

jDC1

n j

!

t^{j 1}Œ^j Œ ^{n j}

! :

Lemma 4.2. LetV_t Dsup¹uWu2PSH.X; Ct /; u0º. Then there exists a con- stantC > 0such that for allt 2.0; 1,

k't VtkL¹.X /C:

Proof. The lemma immediately follows by applying Proposition 1.1 to (4.1).

We now prove the main result in this section.

Lemma 4.3. There exists a constantC > 0such that for allt 2.0; 1, Diam.X; gt/C:

Proof. In this proof we apply a similar argument to that used in the proof of Theo- rem 1.1. Suppose Diam.X; gt/DD for some D6. Let WŒ0; D!X be a smoothing minimizing geodesic with respect to the metricgt and choose the points¹xi D.6i /º^ŒD=6_iD0 . It is clear that the balls¹Bgt.xi; 3/ºare disjoint so

Œ^D₆

X

iD0

Vol.Bgt.xi; 3//

Z

X

DV;

where Vol.Bgt.xi; 3//DR

B_gt.xi; 3/. Hence there exists a geodesic ballBgt.xi; 3/such that

Vol.Bgt.xi; 3//6VD ¹:

We fix such anxi and construct a cut-off function.x/D.r.x//0withr.x/Ddgt.x; xi/ such that

D1 onBgt.xi; 1/; D0 outsideBgt.xi; 2/

and

2Œ0; 1; ¹.⁰/²C; j⁰⁰j C:

Define a functionFt > 0onX such that

Ft D1 outsideBgt.xi; 3/; Ft DD¹² onBgt.xi; 2/

and

C ¹ Z

X

FtC;

Z

X

F_t²C:

We now consider the equation

.Ct C ^t/ⁿDtⁿ e ^tFt for allt 2.0; 1:

Applying Corollary 3.2, there exists a uniform constantC > 0such that for allt 2.0; 1, k ^t VtkL¹.X /C;

and so by Lemma 4.2,

(4.2) k't tkL¹.X /C:

LetgOt DCt t Ci𝜕𝜕 N t. Then onBgt.xi; 2/,

Ric.gOt/D gOtCt ; Ric.gt/D gtCt ; and so

gtlog!Oⁿ_t

!ⁿ_t D nCtrgt.gOt/ nCn !Oⁿ_t

!ⁿ_t ¹_n

: Let

H Dlog!Oⁿ_t

!ⁿ_t :

We may suppose sup_XH DH.z_max/ > 0, otherwise we are done. The pointz_maxmust lie in the support of, and atz_maxwe have

0gtH 1

HgtC2hr;rHi 2H

jrj² n²Cn²e^nH

1

1

2nH² CH

for some uniform constantC > 0for allt 2.0; 1. The maximum principle implies that sup

X

H C.n/I

in particular onBgt.xi; 1/where1, there existsC > 0such that for allt 2.0; 1, O

!ⁿ_t

!ⁿ_t DD¹²e ^{t 't} C:

By the uniformL¹-estimate (4.2), there existsC DC.n; ; ; /such thatDC.

Now we can complete the proof of Theorem 1.2. Gromov’s pre-compactness theorem and the diameter bound in Lemma 4.3 immediately imply that, after passing to a subsequence, .X; gt_j/converges to a compact metric space. Since't Vt is uniformly bounded andVt is uniformly bounded below byV0,'t_j always converges weakly to some'₁ 2PSH.X; /, after passing to a subsequence. In particular, there existsC > 0such that

k'₁ V0kL¹.X / C;

whereV0is the extremal function onX with respect to.

5. Proof of Theorem 1.3 Our proof is based on the arguments of [29, 37, 38].

We fix some notations first. RecallX_can has dimension andis the restriction of the Fubini–Study metric onX_canfrom the embeddingX_can,!CP^Nm, where

NmC1DdimH⁰.X; mKX/:

Henceˆis a smooth nonnegative.1; 1/-form onX, and in the following we identifywith ˆfor simplicity. Letbe a fixed Kähler metric onX.

Define a functionH 2C¹.X /as

^ⁿ DHⁿ

which is the modulus squared of the Jacobian of the mapˆW.X; /!.X_can; /and vanishes on S, the indeterminacy set of ˆ, hence H 2L¹.X; ⁿ/ for some small > 0. We fix a smooth nonnegative function onX_canas defined in [37], which satisfies

0 1; 0p

1𝜕^ N𝜕 C; Ci𝜕𝜕N C;

for some dimensional constantC DC./ > 0. From the construction, vanishes exactly on S⁰Dˆ.S /. There exist > 0,C > 1such that for anyy2X_can^ı DX_cannS⁰(see [37]),

.y/Cinf

Xy

H; hereXyDˆ ¹.y/:

The twisted Kähler–Einstein metricg_t in (1.10) satisfies the following complex Monge–

Ampère equation (with D):

(5.1) .Ct Ci𝜕𝜕'N t/ⁿDtⁿ e^'t for allt 2.0; 1:

In caseKX is semi-ample,Vt D0hence Corollary 3.2 implies (see also [8, 9, 15]):

Lemma 5.1. There is a uniform constantC > 0such thatk'tkL¹.X / C:

We have the following Schwarz lemma whose proof is similar to that of Lemma 2.2, so we omit it.

Lemma 5.2. There exists a constantC > 0such thattr!tC for allt 2.0; 1.

We denotey DjXyfory2X_can^ı , the restriction ofon the fiberXywhich is a smooth .n /-dimensional Calabi–Yau submanifold of X. We will omit the subscript t in 't and simply write' D't, and define

'_y D

«

Xy

'_yⁿ

to be the average of'over the fiberXy. Denote the reference metric!Ot DCt . We calculate .!Ot Ci𝜕𝜕'/N jXy D.t yCi𝜕𝜕.'N '_y//jXy D!tjXy;

hence

.yCt ¹i𝜕𝜕.'N '_y/j^Xy/ⁿ Dt ^nC!ⁿ _t;y : On the other hand,

t ^nC!_t;yⁿ

_yⁿ Dt ^nC!ⁿ _t ^ ⁿ ^ ˇ ˇ ˇ_X

y

C.tr!t/  ⁿ ^

ˇ ˇ ˇ_X

y

CH ¹C .y/:

Since the Sobolev constant of.Xy; y/ is uniformly bounded and the Poincaré constant of .Xy; y/is bounded byC e^{B .y/}for some uniform constantsB; C > 0(see [37]), combined

with the fact that «

Xy

.' '_y/_yⁿ D0;

Moser iteration implies ([37, 40]):

Lemma 5.3. There exist constantsB₁; C₁> 0such that for anyy2X_can^ı , sup

Xy

t ¹j' '_yj C1e^{B1 .y/} for allt 2.0; 1:

Proposition 5.1. On any compact subsetK bXnS, there existsC DC.K/ > 1such that for allt 2.0; 1,

C ¹!Ot !t C!Ot onK:

Given theC⁰-estimate in Lemma 5.3, Proposition 5.1 can be proved by theC²-estimate ([40]) for the Monge–Ampère equation together with a modification as in [29, 37, 38], so we omit the proof.

Let us recall the construction of the canonical metric !_can on X_can^ı (see [29]). Define a functionF D ^ˆ onX_can^ı , andF is inL^1C"for some small" > 0([29]). The metric!_can is obtained by solving the following complex Monge–Ampère equation onX_can:

.Ci𝜕𝜕'N ₁/ D n

!

F e^'1

for '₁ 2PSH.X_can; /\C⁰.X_can/\C¹.X_can^ı /. Then !_canDCi𝜕𝜕'N ₁, and in the fol- lowing we will write₁ D!_can.