Infinite time singularities of the Kähler-Ricci flow
aa r X i v : . [ m a t h . DG ] D ec INFINITE TIME SINGULARITIES OF THEK ¨AHLER-RICCI FLOW
VALENTINO TOSATTI AND YUGUANG ZHANG
Abstract.
We study the long-time behavior of the K¨ahler-Ricci flow oncompact K¨ahler manifolds. We give an almost complete classificationof the singularity type of the flow at infinity, depending only on theunderlying complex structure. If the manifold is of intermediate Kodairadimension and has semiample canonical bundle, so that it is fiberedby Calabi-Yau varieties, we show that parabolic rescalings around anypoint on a smooth fiber converge smoothly to a unique limit, whichis the product of a Ricci-flat metric on the fiber and of a flat metricon Euclidean space. An analogous result holds for collapsing limits ofRicci-flat K¨ahler metrics. Introduction
This paper has two main goals. The first goal is to prove some resultsabout collapsing limits of either Ricci-flat K¨ahler metrics or certain solutionsof the K¨ahler-Ricci flow, which complement and complete the very recentresults proved by Weinkove, Yang and the first-named author in [28]. In anutshell, we prove higher-order a priori estimates for the certain rescalingsof the solutions restricted to the fibers of a holomorphic map, and we thenclassify the possible blowup limit spaces. As it turns out, these limits areunique, and are given by the product of a Ricci-flat K¨ahler metric on acompact Calabi-Yau manifold times a Euclidean metric on C m .The second goal of the paper is to study infinite time singularities of theK¨ahler-Ricci flow. As we will recall below, these are divided into two classescalled “type IIb” and “type III” (the other cases “type I” and “type IIa” arereserved for finite-time singularities). Assuming that the canonical bundleis semiample (which conjecturally is always true whenever we have a long-time solution), we give an almost complete classification (which is completein complex dimension 2) of which singularity type arises, depending on thecomplex structure of the underlying manifold. As a consequence, in all thecases that we cover, the singularity type does not depend on the initialK¨ahler metric.We now discuss the first goal in detail. As we said above, we will considertwo different setups. In the first setup, which is the same as in [25, 8, 9, 11, The first-named author is supported in part by NSF grant DMS-1308988 and by a SloanResearch Fellowship. The second-named author is supported in part by grant NSFC-11271015.
X, ω X ) be a compact Ricci-flat Calabi-Yau ( n + m )-manifold, n >
0, which admits a holomorphic map π : X → Z where ( Z, ω Z ) is acompact K¨ahler manifold. Denote by B = π ( X ) the image of π , and assumethat B is an irreducible normal subvariety of Z with dimension m >
0, andthat the map π : X → B has connected fibers. Denote by χ = π ∗ ω Z , whichis a smooth nonnegative real (1 , X whose cohomology class lies onthe boundary of the K¨ahler cone of X , and denote also by χ the restrictionof ω Z to the regular part of B . In practice we can take Z = CP N if B is a projective variety, or Z = B if B is smooth. In general, given a map π : X → B as above, there is a proper analytic subvariety S ′ ⊂ B , whichconsists of the singular points of B together with the critical values of π ,such that if we let S := π − ( S ′ ) we obtain an analytic subvariety of X and π : X \ S → B \ S ′ is a smooth submersion. For any y ∈ B \ S ′ the fiber X y = π − ( y ) is a smooth Calabi-Yau manifold of dimension n , and it isequipped with the K¨ahler metric ω X | X y . Consider the K¨ahler metrics on X given by ˜ ω = ˜ ω ( t ) = χ + e − t ω X , with t >
0, and call ω = ω ( t ) = ˜ ω + √− ∂∂ϕ the unique Ricci-flat K¨ahler metric on X cohomologous to ˜ ω , which existthanks to Yau’s Theorem [30], with potentials normalized by sup X ϕ = 0.They satisfy a family of complex Monge-Amp`ere equations(1.1) ω n + m = (˜ ω + √− ∂∂ϕ ) n + m = c t e − nt ω n + mX , where c t is a constant that has a positive limit as t → ∞ . Thanks to theworks [25, 21, 8, 9, 28, 32] we know that in this case the Ricci-flat metrics ω ( t ) have bounded diameter and collapse locally uniformly on X \ S to acanonical K¨ahler metric on B \ S ′ , and when the fibers X y are tori then thecollapsing is smooth and with locally bounded curvature on X \ S . Also, forgeneral smooth fibers, the rescaled metrics along the fibers e t ω | X y convergein C α to a Ricci-flat metric on X y .The second setup is as follows (cf. [21, 22, 23, 5, 28]). Now ( X, ω X ) is acompact K¨ahler ( n + m )-manifold, n >
0, with semiample canonical bundleand Kodaira dimension equal to m >
0. As explained for example in [28],sections of K ℓX , for ℓ large, give rise to a fiber space π : X → B called the Iitaka fibration of X , with B a normal projective variety of dimension m and the smooth fibers X y = π − ( y ) , y ∈ B \ S ′ also Calabi-Yau n -manifolds.We let χ be the restriction of ℓ ω F S to B , as well as its pullback to X . Thistime we consider the solution ω = ω ( t ) of the normalized K¨ahler-Ricci flow(1.2) ∂∂t ω = − Ric( ω ) − ω, ω (0) = ω X , which exists for all t >
0. Thanks to [21, 22, 23, 5, 28] we have thatthe evolving metrics have uniformly bounded scalar curvature and collapselocally uniformly on X \ S to a canonical K¨ahler metric on B \ S ′ , and againwe have smooth collapsing when the smooth fibers are tori. Again, therescaled metrics along the fibers e t ω | X y converge in C α to a Ricci-flat metricon X y . NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 3
Our first goal is to improve this last statement by showing the followinghigher-order estimates:
Theorem 1.1.
Assume we are in either the first or second setup. Given acompact subset K ⊂ B \ S ′ and k > there is a constant C k such that (1.3) (cid:13)(cid:13) e t ω | X y (cid:13)(cid:13) C k ( X y ,ω X | Xy ) C k , e t ω | X y > C − ω X | X y , holds for all t > and for all y ∈ K . The case when k = 1 of Theorem 1.1 was proved recently in [28], whoalso showed that as t → ∞ the metrics e t ω | X y converge in C α ( X y , ω X | X y ) , < α <
1, to the unique Ricci-flat K¨ahler metric on X y cohomologous to ω X | X y . Combining this and Theorem 1.1 we immediately conclude: Corollary 1.2.
Assume we are in either the first or second setup. Given y ∈ B \ S ′ we have (1.4) e t ω | X y → ω SRF,y , in the smooth topology on X y as t → ∞ , where ω SRF,y is the unique Ricci-flatmetric on X y cohomologous to ω X | X y . This solves affirmatively a problem raised by the first-named author [26,Question 4.1], [27, Question 3].In the second setup we investigate in more detail the nature of the sin-gularity of the K¨ahler-Ricci flow as t → ∞ , and more precisely we wantto determine the possible limits (or “tangent flows”) that one obtains byparabolically rescaling the flow around the “singularity at infinity”. In gen-eral determining whether tangents to solutions of geometric PDEs are uniqueor not is a very challenging problem. In the setup as above, we prove thatsuch parabolic rescalings (centered at a point on a smooth fiber) convergeto a unique limit. More precisely we have: Theorem 1.3.
Assume the same setup for the K¨ahler-Ricci flow (1.2) asabove. Given t k → ∞ and x ∈ X \ S , let V be the preimage of a sufficientlysmall neighborhood of π ( x ) and let ω k ( t ) = e t k ω ( te − t k + t k ) be the parabolically rescaled flows, then after passing to a subsequence theflows ( V, ω k ( t ) , x ) , t ∈ [ − , , converge in the smooth Cheeger-Gromov senseto ( X y × C m , ω ∞ , ( x, , where y = π ( x ) , and ω ∞ is the product of the uniqueRicci-flat K¨ahler metric on X y in the class [ ω X | X y ] and of a flat metric on C m , viewed as a static solution of the flow. In particular, there is a uniquesuch limit up to holomorphic isometry. In fact, analogous results as in Theorem 1.3 also hold in the first setupof collapsing of Calabi-Yau manifolds, with essentially the same proofs. Webriefly discuss this at the end of Section 3.
V. TOSATTI AND Y. ZHANG
We now come to the second goal of this paper, which is to characterizeinfinite time singularities of the K¨ahler-Ricci flow. Recall that a long-timesolution of the unnormalized
K¨ahler-Ricci flow(1.5) ∂∂t ω = − Ric( ω ) , ω (0) = ω X , is called type IIb if sup X × [0 , ∞ ) t | Rm( ω ( t )) | ω ( t ) = + ∞ , and type III if sup X × [0 , ∞ ) t | Rm( ω ( t )) | ω ( t ) < + ∞ . If the flow is instead normalized as in (1.2), then one has to remove thefactor of t from these conditions.When dim X = 1, so X is a compact Riemann surface, it follows easilyfrom the work of Hamilton [10] that all long-time solutions of the flow are oftype III, and these are exactly the K¨ahler-Ricci flow solutions on compactRiemann surfaces of genus g >
1. Recent work of Bamler [1] shows that thesame statement holds for the Ricci flow (not K¨ahler) on compact Riemannian3-manifolds. Homogeneous type-III solutions were studied by Lott [15]. Thesimplest example of a type IIb solution on a compact Riemannian 4-manifoldis a non-flat Ricci-flat K¨ahler metric on a K π : X → B is a holomorphic submersion with fibers equal to complex tori,with c ( B ) <
0, with X projective and initial K¨ahler class rational, thenthe flow is of type III. This used key ideas of Gross-Tosatti-Zhang [8] inthe case of collapsing of Ricci-flat metrics, and the projectivity/rationalityassumptions were recently removed in [11]. See also [7] for the case when X is a product.Our goal is to have a more detailed understanding of which long-timesolutions of the K¨ahler-Ricci flows are of type IIb or type III. Our main toolis the following observation. Proposition 1.4.
Let X be a compact K¨ahler manifold with K X nef andwhich contains a possibly singular rational curve C ⊂ X (i.e. C is thebirational image of a nonconstant holomorphic map f : CP → X ) such that R C c ( X ) = 0 . Then any solution of the unnormalized K¨ahler-Ricci flow (1.5) on X must be of type IIb. For example if X is a minimal K¨ahler surface of general type which con-tains a ( − NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 5
Now in general if a K¨ahler-Ricci flow solution on a compact K¨ahler man-ifold X exists for all t >
0, then necessarily the canonical bundle K X isnef (the converse is also true [24]). The abundance conjecture in algebraicgeometry (or rather its generalization to K¨ahler manifolds) predicts that if K X is nef then in fact it is semiample, namely K ℓX is base-point-free forsome ℓ >
1. We will assume that this is the case, and so the sections of K ℓX ,some ℓ large, define a fiber space π : X → B , the Iitaka fibration of X . Asbefore we denote by S ′ ⊂ B the singular set of B together with the criticalvalues of π , and let S = π − ( S ′ ), so that S = ∅ precisely when B is smoothand π is a submersion. Somewhat imprecisely, we will refer to S as the setof singular fibers of π . We will also write X y = π − ( y ) for one of the smoothfibers, y ∈ B \ S ′ . We then have the following result. Theorem 1.5.
Let X be a compact K¨ahler n -manifold with K X semiample,and consider a solution of the K¨ahler-Ricci flow (1.5) . • κ ( X ) = 0 ⋄ X is not a finite quotient of a torus ⇒ Type IIb ⋄ X is a finite quotient of a torus ⇒ Type III • κ ( X ) = n ⋄ K X is ample ⇒ Type III ⋄ K X is not ample ⇒ Type IIb • < κ ( X ) < n ⋄ X y is not a finite quotient of a torus ⇒ Type IIb ⋄ X y is a finite quotient of a torus and S = ∅ ⇒ Type IIIIn particular, in these cases the type of singularity does not depend on theinitial metric.
This result leaves out only the case when 0 < κ ( X ) < n and the generalfiber X y is a finite quotient of a torus and there are singular fibers. Inthis case, sometimes one can find a component of a singular fiber which isuniruled, in which case the flow is type IIb by Proposition 1.4, but in someother cases there is no such component, and then it seems highly nontrivialto determine whether the flow is of type III or IIb. In any case, we expectthe singularity type to be always independent of the initial metric.We are able to solve this question when n = 2, and complete the sin-gularity classification (note that since the abundance conjecture holds forsurfaces, it is enough to assume that K X is nef). In this case π : X → B isan elliptic fibration with X a minimal properly elliptic K¨ahler surface, andwith some singular fibers. Recall that in Kodaira’s terminology (cf. [2]) afiber of type mI , m > m . Wecan then complete Theorem 1.5 in dimension 2 as follows: Theorem 1.6.
Let X be a minimal K¨ahler surface with κ ( X ) = 1 , and let π : X → B be an elliptic fibration which is not a submersion everywhere.Then the flow is type III if and only if the only singular fibers of π are oftype mI , m > . V. TOSATTI AND Y. ZHANG
In the Minimal Model Program, we sometime have many different min-imal models in one birational equivalence class. In [13], Kawamata showsthat different minimal models can be connected by a sequence of flops. Acorollary of Proposition 1.4 is the relationship of this multi-minimal modelphenomenon and the singularity type of the K¨ahler-Ricci flow.
Corollary 1.7.
Let X be a projective n -manifold with K X nef. If X has adifferent minimal model Y , i.e. a Q -factorial terminal variety Y with K Y nef and with a birational map α : Y X , which is not isomorphic to X ,then any solution of the K¨ahler-Ricci flow (1.5) on X is of type IIb. Asa consequence, if there is a flow solution of type III, then X is the uniqueminimal model in its birational equivalence class. This paper is organized as follows. In Section 2 we prove Theorem 1.1.Theorem 1.3 is proved in Section 3, and finally in Section 4 we give theproofs of Proposition 1.4, Theorems 1.5 and 1.6 and Corollary 1.7.
Acknowledgements:
We are grateful to M. Gross, H.-J. Hein, B. Weinkovefor very useful discussions. Most of this work was carried out while the first-named author was visiting the Mathematical Science Center of TsinghuaUniversity in Beijing, which he would like to thank for the hospitality.2.
Estimates along the fibers
In this section we prove Theorem 1.1 by deriving a priori C ∞ estimatesfor the rescaled metrics restricted to a smooth fiber.We first consider the setup of collapsing of Ricci-flat K¨ahler metrics, asdescribed in the Introduction. We need two preliminary results. The firstone follows from work of the first-named author [25] (see also [8, Lemma4.1]). Lemma 2.1 (see [25, 8]) . Given any compact set K ⊂ X \ S there is aconstant C such that on K the Ricci–flat metrics ω satisfy (2.1) C − ( χ + e − t ω X ) ω C ( χ + e − t ω X ) , for all t > . The second ingredient is the following local estimate for Ricci-flat K¨ahlermetrics, which is contained in [11, Sections 3.2 and 3.3] and is an adaptation(and in fact a special case) of a similar result from [20] for the K¨ahler-Ricciflow.
Lemma 2.2 (see [11]) . Let B (0) be the unit ball in C n and let ω E be theEuclidean metric. Assume that ω is a Ricci-flat K¨ahler metric on B (0) which satisfies (2.2) A − ω E ω A ω E , NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 7 for some positive constant A . Then for any k > there is a constant C k that depends only on k, n, m, A such that on B / (0) we have (2.3) k ω k C k ( B / (0) ,ω E ) C k . Proof of Theorem 1.1 for Ricci-flat K¨ahler metrics.
Fix a point y ∈ B \ S ′ ,a point x ∈ X y and a small chart U ⊂ X with coordinates ( y, z ) =( y , . . . , y m , z , . . . , z n ) centered at x , where ( y , . . . , y m ) are the pullbackvia π of coordinates on the image π ( U ) ⊂ B and such that the projection π in this coordinates is just ( y, z ) y . Such a chart exists because π isa submersion near x (see [14, p.60]). We can assume that U equals thepolydisc where | y i | < i m and | z α | < α n .For each t >
0, consider the polydiscs B t = B e t/ (0) ⊂ C m , let D be theunit polydisc in C n and define maps F t : B t × D → U, F t ( y, z ) = ( ye − t/ , z ) . Note that the stretching F t is the identity when restricted to { } × D , andas t approaches zero the polydiscs B t × D exhaust C m × D . On U we canwrite ω X ( y, z ) = √− m X i,j =1 g ij ( y, z ) dy i ∧ dy j + 2Re √− m X i =1 n X α =1 g iα ( y, z ) dy i ∧ dz α + √− n X α,β =1 g αβ ( y, z ) dz α ∧ dz β , so that F ∗ t ω X ( y, z ) = e − t √− m X i,j =1 g ij ( ye − t/ , z ) dy i ∧ dy j + 2 e − t/ Re √− m X i =1 n X α =1 g iα ( ye − t/ , z ) dy i ∧ dz α + √− n X α,β =1 g αβ ( ye − t/ , z ) dz α ∧ dz β . Clearly as t goes to infinity, these metrics converge smoothly on compactsets of C m × D to the nonnegative form η = √− P nα,β =1 g αβ (0 , z ) dz α ∧ dz β ,which is constant in the y directions and is just equal to the restriction of ω X on the fiber X y ∩ U , under the identification X y ∩ U = { } × D . Therescaled pullback metrics e t F ∗ t ω ( t ) are Ricci-flat K¨ahler on B t × D . Next,we pull back equation (2.1) and multiply it by e t and get(2.4) C − (cid:0) e t F ∗ t χ + F ∗ t ω X (cid:1) e t F ∗ t ω C (cid:0) e t F ∗ t χ + F ∗ t ω X (cid:1) , V. TOSATTI AND Y. ZHANG which holds on B t × D . But χ is the pullback of a metric √− m X i,j =1 χ ij ( y ) dy i ∧ dy j on B and so we have that e t ( F ∗ t χ )( y, z ) = √− m X i,j =1 χ ij ( ye − t/ ) dy i ∧ dy j , which as t goes to infinity converge smoothly on compact sets of C m × D to the nonnegative form η ′ = √− P mi,j =1 χ ij (0) dy i ∧ dy j . In particular, η + η ′ is a K¨ahler metric on C m × D which is uniformly equivalent to ω E on the whole of C m × D . Since η + η ′ is the limit as t goes to infinity of e t F ∗ t χ + F ∗ t ω X , we see that given any compact set K of B t × D there is aconstant C independent of t such that on K we have(2.5) C − ω E e t F ∗ t ω Cω E . Using (2.5), we can then apply Lemma 2.2 to get that the Ricci-flat metrics e t F ∗ t ω have uniform C ∞ bounds on any compact subset of B t × D . If wenow restrict to { } × D , which is identified with X y ∩ U , then the maps F t are just the identity, and by covering X y by finitely many such charts thisshows that the rescaled metrics along any smooth fiber e t ω | X y have uniform C ∞ ( X y , ω X | X y ) bounds. The uniform lower bound for e t ω | X y follows atonce from (2.1).Finally, the fact that the estimates are uniform as we vary y in a compactsubset of B \ S ′ follows from the fact that all the constants in the proof wejust finished vary continuously as we vary y . (cid:3) Next, we consider the second setup from the Introduction, of collapsing ofthe K¨ahler-Ricci flow. In this case the same estimate as in (2.1) was provedin [5] (see also [21]), and the local estimates that replace Lemma 2.2 aregiven in [20].
Proof of Theorem 1.1 for the K¨ahler-Ricci flow.
Given x ∈ X \ S , y = π ( x ),and t k → ∞ , we will show that there are constant C ℓ , ℓ > , such that(2.6) (cid:13)(cid:13) e t k ω ( t k ) | X y (cid:13)(cid:13) C ℓ ( X y ,ω X | Xy ) C ℓ , e t k ω ( t k ) | X y > C − ω X | X y , for all k, ℓ >
0, and that these estimates are uniform as y varies in a compactset of B \ S ′ . Once this is proved, the estimates stated in Theorem 1.1 followeasily from an argument by contradiction, using compactness.As in the case of Ricci-flat K¨ahler metrics, we choose a chart U centered at x with local product coordinates ( y, z ) as before, with y in the unit polydiscin C m and z in the unit polydisc D in C n , and define stretching maps F k : B k × D → U, F k ( y, z ) = ( ye − t k / , z ) , NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 9 where B k = B e tk/ (0) ⊂ C m . Thanks to the analog of Lemma 2.1 in [5], on U we have C − ( χ + e − t ω X ) ω ( t ) C ( χ + e − t ω X ) . We consider the parabolically rescaled and stretched metrics˜ ω k ( t ) = e t k F ∗ k ω ( te − t k + t k ) , t ∈ [ − , . On U we have that C − ( e t k χ + e − te − tk ω X ) e t k ω ( te − t k + t k ) C ( e t k χ + e − te − tk ω X ) , and since we assume that t ∈ [ − , C − ( e t k χ + ω X ) e t k ω ( te − t k + t k ) C ( e t k χ + ω X ) , and so(2.7) C − F ∗ k ( e t k χ + ω X ) ˜ ω k ( t ) CF ∗ k ( e t k χ + ω X ) . The same calculation as in the case of Ricci-flat metrics shows that themetrics F ∗ k ( e t k χ + ω X ) converge smoothly on compact sets of C m × D to theproduct of a flat metric on C m and the metric ω X | X y ∩ D , and in particulargiven any compact subset K of B k × D there is a constant C such that on K we have C − ω E F ∗ k ( e t k χ + ω X ) Cω E , where ω E is the Euclidean metric on C n + m . Therefore on K × [ − ,
0] wehave C − ω E ˜ ω k ( t ) Cω E . The metrics ˜ ω k ( t ) for t ∈ [ − ,
0] satisfy(2.8) ∂∂t ˜ ω k ( t ) = − Ric(˜ ω k ( t )) − e − t k ˜ ω k ( t ) , and note that the coefficient e − t k is uniformly bounded (and in fact goes tozero). Then the interior estimates of [20] give us that, up to shrinking K slightly, k ˜ ω k ( t ) k C ℓ ( K,ω E ) C ℓ , ˜ ω k ( t ) > C − ω E , for t ∈ [ − / ,
0] and for all k, ℓ , and for some uniform constants C ℓ . Setting t = 0 we obtain local C ∞ bounds for the metrics e t k F ∗ k ω ( t k ). If we nowrestrict to { } × D , which is identified with X y ∩ U , then the maps F k are just the identity, and by covering X y by finitely many such charts thisproves (2.6). Again, the fact that the estimates are uniform as y varies ina compact subset of B \ S ′ follows from the proof we just finished. (cid:3) Blowup limits of the K¨ahler-Ricci flow
In this section we describe the possible blowup limits at time infinity ofthe K¨ahler-Ricci flow in the same setup as in the Introduction, thus provingTheorem 1.3.
Proof of Theorem 1.3.
Given a point x ∈ X \ S , y = π ( x ), and t k → ∞ ,choose a chart U centered at x with local product coordinates ( y, z ) asbefore, where z varies in the unit polydisc D ⊂ C n and y in the unit polydiscin C m . We let ω k ( t ) = e t k ω ( te − t k + t k ) , be the parabolic rescalings of the metrics along the flow, with t ∈ [ − , F k : B k × D → U, F k ( y, z ) = ( ye − t k / , z ) , where B k = B e tk/ (0) ⊂ C m , and we have that the metrics F ∗ k ω k ( t ) , t ∈ [ − ,
0] have uniform C ∞ bounds on compact sets of C m × D , and therefore upto passing to a subsequence they converge smoothly to a solution ω ∞ ( t ) , t ∈ [ − ,
0] of the K¨ahler-Ricci flow ∂∂t ω ∞ ( t ) = − Ric( ω ∞ ( t )) , on C m × D . This limit flow is unnormalized because the coefficient e − t k in(2.8) converges to zero. Also, passing to the limit in (2.7) we see that(3.1) C − ( ω X | X y ∩ D + ω E ) ω ∞ ( t ) C ( ω X | X y ∩ D + ω E ) , on the whole of C m × D and for all t >
0, where ω E is a flat metric on C m .However, the original metrics ω ( t ) have a uniform bound on their scalarcurvature thanks to [23], so after rescaling the scalar curvature of ˜ ω k ( t ) goesto zero uniformly as k goes to infinity. Therefore ω ∞ ( t ) is scalar flat, andfrom the pointwise evolution equation for its scalar curvature, we see that ω ∞ ( t ) is Ricci-flat, and hence independent of t .Out next task is to glue together these local limits to obtain a globalCheeger-Gromov limit on X y × C m . To do this, we cover X y by finitelymany charts { U α } as above, and let V = ∪ α U α . On each U α we have localproduct coordinates ( y, z α ), with y and z α belonging to the unit polydiscsin C m and C n respectively, and π is given by ( y, z α ) y . If U α ∩ U β = ∅ ,then on this overlap the two local coordinates are related by holomorphictransformation maps z β = h αβ ( y, z α ) . Fix a radius
R > B k = B e − tk/ R (0) be the polydisc of radius e − t k / R , where k is large enough so that e − t k / R <
1. We define a holo-morphic coordinate change on ˜ B k by y k = e t k / y, so that ˜ B k becomes thepolydisc B R (0) of radius R in these new coordinates. This is the same asapplying the stretching maps F k as earlier. Then the complex manifold NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 11 V k = π − ( ˜ B k ) (here we are viewing ˜ B k as a subset of B ) is obtained bygluing the polydiscs { ( w, z α ) | w ∈ B R (0) , z α ∈ D } , { ( w, z β ) | w ∈ B R (0) , z β ∈ D } via the transformation maps z β = h αβ ( we − t k / , z α ) . As k → ∞ , these converge smoothly on compact sets to the transformationmaps z β = h αβ (0 , z α ) , which give the product manifold B R (0) × X y . Also, the metrics e t k ( χ + e − t k ω X ) on V k after changing coordinates converge smoothly on compactsets as k → ∞ to the product metric ω E + ω X | X y where ω E is a Euclideanmetric on C m , as in the proof of Theorem 1.1. On the other hand, asdiscussed above, the metrics ω k ( t ) after coordinate change live on V k andafter passing to a subsequence converge smoothly on compact sets to a Ricci-flat K¨ahler metric ω ∞ (independent of t ) on B R (0) × X y . By making R larger and larger, we obtain the desired Cheeger-Gromov convergence of theflow on V to a Ricci-flat K¨ahler metric ω ∞ on X y × C m . Also, thanks to(3.1), we have that ω ∞ is uniformly equivalent to ω E + ω X | X y on X y × C m .It remains to show that ω ∞ is the product of a Ricci-flat K¨ahler metricon X y and of a flat metric on C m . Thanks to the recent results in [28,(1.8)] we know that the restrictions ω ∞ | X y , y ∈ C m , are all equal to thesame Ricci-flat K¨ahler metric ω F := ω SRF,y on X y in the class [ ω X | X y ].For simplicity of notation, let us also write F = X y .We claim that there is a smooth function u on F × C m such that(3.2) ω ∞ = ω F + ω E + √− ∂∂u, on F × C m , where ω F + ω E is the product Ricci-flat metric. To prove (3.2),recall [22] that the K¨ahler-Ricci flow is of the form ω ( t ) = (1 − e − t ) χ + e − t ω X + √− ∂∂ϕ ( t ) , for some potentials ϕ ( t ). Therefore on B k × D we can write(3.3) ˜ ω k ( t ) = F ∗ k ω k ( t ) = ω ref ,k ( t ) + √− ∂∂ ( e t k F ∗ k ϕ ( te − t k + t k )) , where as mentioned above the reference metrics ω ref ,k ( t ) = e t k F ∗ k (cid:16) (1 − e − te − tk − t k ) χ + e − te − tk − t k ω X (cid:17) , converge smoothly on compact sets to the product metric ω E + ω X | F (inde-pendent of t ), as k → ∞ .We observe that [ ω ∞ ] = [ ω F + ω E ] in H ( F × C m ), and so there is a real1-form ζ on X × C m such that(3.4) ω ∞ = ω F + ω E + dζ = ω F + ω E + ∂ζ , + ∂ζ , , ∂ζ , = 0 , where ζ = ζ , + ζ , . Since c ( F ) = 0, the Bogomolov-Calabi decompositiontheorem shows that there is a finite unramified cover π : T × ˜ F × C m → F × C m , where T is a torus (possibly a point), and ˜ F is simply connected with c ( ˜ F ) = 0 (also possibly a point). The Leray spectral sequence computingthe Dolbeault cohomology of the product T × ˜ F × C m degenerates at thefirst page, giving H , ( T × ˜ F × C m ) ∼ = H , ( T ) ⊗ H ( C m , O C m ) , where we used that H , ( C m ) = 0 by the ∂ -Poincar´e Lemma. If we write T = C k / Λ, and let { z i } be the standard coordinates on C k , then H , ( T ) isgenerated by the constant coefficient forms { dz i } , and so on T × ˜ F × C m wecan write π ∗ ζ , = k X i =1 σ i ( y ) dz i + ∂h, for some holomorphic functions σ i ( y ) on C m and a complex-valued function h , where here dz i also denote their pullbacks to T × ˜ F × C m . Therefore π ∗ ω ∞ = π ∗ ( ω F + ω E ) + 2Re X i dσ i ∧ d ¯ z i + 2 √− ∂∂ Im h. Note that ω F is ∂∂ -cohomologous to ω X | F on F , and hence ω F = ω X | F + √− ∂∂v for a smooth function v . If we denote by u k = π ∗ ( e t k F ∗ k ϕ ( te − t k + t k ) − v ) − h, then by (3.3) we have that, √− ∂∂u k → π ∗ ω ∞ − π ∗ ( ω F + ω E ) − √− ∂∂ Im h = 2Re X i dσ i ∧ d ¯ z i , smoothly on compact subsets of T × ˜ F × C m . In particular, restricting thisto any fiber T × ˜ F × { y } , y ∈ C m , we see that √− ∂∂u k | T × ˜ F ×{ y } → , smoothly as k → ∞ . If we let u k be the smooth function on C m obtained byaveraging u k on these fibers, with respect to π ∗ ω nF , then √− ∂∂u k also hasuniform local C ∞ bounds. The functions u k − u k thus have fiberwise averagezero and the forms √− ∂∂ ( u k − u k ) have uniform C ∞ bounds on compactsets and their fiberwise restrictions go to zero smoothly, and therefore u k − u k converge to zero locally uniformly, and hence locally smoothly on T × ˜ F × C m . It follows that √− ∂∂ ( u k − u k ) → P i dσ i ∧ d ¯ z i , which is the limit of √− ∂∂u k , is also equal to thelimit of √− ∂∂u k . But since √− ∂∂u k are forms on C m , this implies that2Re P i dσ i ∧ d ¯ z i is also the pullback of a form on C m , which is only possibleif dσ i = 0 for all i . This gives π ∗ ∂ζ , = ∂∂h. If we let ˜ h be the average of h under the action of the (finite) deck transfor-mation group of π , then ˜ h descends to a complex-valued function on F × C m NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 13 and we have ∂ζ , = ∂∂ ˜ h, which together with (3.4) proves (3.2) with u = 2Im˜ h . This argumentis similar to [8, Proposition 3.1], but we have exploited the more specialsetup here to obtain (3.2) without the need of passing to a holomorphic“translation” in the T factor, which is needed in general.Now that we have (3.2), we can restrict it to any fiber F ×{ y } , and since wehave that ω ∞ | F ×{ y } = ω F for all y , we conclude that ( √− ∂∂u ) | F ×{ y } = 0,i.e. u | F ×{ y } is a constant (which depends on y ). In other words, u is thepullback of a smooth function on C m . Therefore (3.2) now says that ω ∞ is the product of the Ricci-flat K¨ahler metric ω F on F times the K¨ahlermetric ˆ ω := ω E + √− ∂∂u on C m . Since ω ∞ is Ricci-flat, we have that ˆ ω is Ricci-flat as well, and the only thing left to prove is that ˆ ω is flat. Butearlier we proved that C − ( ω F + ω E ) ω ∞ C ( ω F + ω E ) , on the whole of F × C m , and so we conclude that C − ω E ˆ ω Cω E , on C m . We can write ˆ ω = √− ∂∂ϕ on C m and the function log det( ϕ ij ) isharmonic and bounded on C m , hence constant, and we can then apply [19,Theorem 2] to conclude that ˆ ω is flat. (cid:3) In the end of the proof we used a Liouville-type theorem that a Ricci-flatK¨ahler metric on C m which is uniformly equivalent to the Euclidean metricmust be flat. The proof in [19] uses integral estimates, but in fact this resultcan also be easily proved using a local Calabi-type C estimate as in [30](we leave the details to the interested reader). Another related Liouvilletheorem was proved recently in [29].Lastly, we mention the analogous result as Theorem 1.3 for the case ofRicci-flat K¨ahler metrics (the first setup in the Introduction). Theorem 3.1.
Assume the first setup in the Introduction, as in (1.1) .Given x ∈ X \ S , let V be the preimage of a sufficiently small neighbor-hood of π ( x ) . Then ( V, e t ω ( t ) , x ) , converge in the smooth Cheeger-Gromovsense as t → ∞ to ( X y × C m , ω ∞ , ( x, , where y = π ( x ) , and ω ∞ is theproduct of the unique Ricci-flat K¨ahler metric on X y in the class [ ω X | X y ] and of a flat metric on C m .Proof. Since the proof is almost identical to the one of Theorem 1.3, we willbe very brief. First we work on a small polydisc centered at x with localproduct coordinates, and as in the proof of Theorem 1.1 we show that afterstretching the coordinates the metrics e t ω ( t ) have uniform C ∞ bounds, andtherefore we obtain sequential limits which are Ricci-flat K¨ahler metrics. Asin the proof of Theorem 1.3 limits on different charts glue together to givea Ricci-flat K¨ahler metric on X y × C m , which is uniformly equivalent to aproduct metric. The same argument as before shows that such a metric is unique up to holomorphic isometry, and is the product of a Ricci-flat metricon X y and a flat metric on C m . It follows that this is the Cheeger-Gromovlimit of the whole family e t ω ( t ), without passing to subsequences. (cid:3) Infinite time singularities of the K¨ahler-Ricci flow
In this section we study the singularity types of long-time solutions of theK¨ahler-Ricci flow.We wish to prove the criterion stated in Proposition 1.4, which ensuresthat a long-time solution of the unnormalized K¨ahler-Ricci flow (1.5) is oftype IIb. First, we make an elementary observation: if there are points x k ∈ X , times t k → ∞ , 2-planes π k ⊂ T x k X and a constant κ > ω ( t k ) ( π k ) > κ, for all k , then in particular sup X | Rm( ω ( t k )) | ω ( t k ) > κ, and sosup X × [0 , ∞ ) t | Rm( ω ( t )) | ω ( t ) = + ∞ , and the flow is type IIb.We now consider Proposition 1.4. To give an intuition for why such aresult should hold, let us first assume that the map f : CP → X is asmooth embedding. Then we can apply the Gauß-Bonnet theorem to get4 π = Z CP K ( f ∗ ω ( t )) f ∗ ω ( t ) sup CP K ( f ∗ ω ( t )) Z C ω ( t ) sup C K ( ω ( t )) Z C ω X , where K ( f ∗ ω ( t )) denotes the Gauß curvature of the pullback metric, whilesup C K ( ω ( t )) is the maximum of the bisectional curvatures of ω ( t ) at pointsof C ⊂ X . Here we used the facts that the bisectional curvature decreasesin submanifolds, and that R C c ( X ) = 0. We can therefore apply the obser-vation above and conclude that the flow is type IIb. We now give the proofin the general case. Proof of Proposition 1.4.
Let ω ( t ) be any solution of the unnormalized K¨ahler-Ricci flow (1.5). Since K X is nef, we know that ω ( t ) exists for all positive t .Note that the existence of a rational curve C in X implies that X does notadmit any K¨ahler metric with nonpositive bisectional curvature, by Yau’sSchwarz Lemma [31]. In particular K ( t ) = sup X Bisec ω ( t ) , the largest bisec-tional curvature of ω ( t ) , satisfies K ( t ) > t . Our goal is to give aneffective uniform positive lower bound for K ( t ). We now work on C ⊂ CP ,so that we have a nonconstant entire holomorphic map f : C → X . Considerthe time-dependent smooth function e ( t ) on C (the “energy density”) givenby e ( t ) = tr ω E ( f ∗ ω ( t )) , where ω E is the Euclidean metric on C . The usual Schwarz Lemma calcu-lation [31] gives ∆ E e ( t ) > − K ( t ) e ( t ) . NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 15
Since the map f is nonconstant, there is one point in C where e ( t ) is nonzerofor all t , and we may assume that this is the origin. A standard ε -regularityargument (see e.g. [16, Lemma 4.3.2]) shows that for each fixed t if(4.2) Z B r e ( t ) π K ( t ) , then we have(4.3) e ( t )(0) πr Z B r e ( t ) , where B r is the Euclidean disc of radius r centered at the origin. Thanks tothe assumption that R C c ( X ) = 0, we havelim r →∞ Z B r e ( t ) = Z C ω ( t ) = Z C ω X , which is a positive constant. For a fixed t , if (4.2) was true for all r > r → ∞ in (4.3) and obtain e ( t )(0) = 0, a contradiction.Therefore, for each t there is some r ( t ) > π K ( t ) Z B r ( t ) e ( t ) Z C ω X , and so K ( t ) > κ >
0. Since each bisectional curvature is the sum of twosectional curvatures, we conclude that ω ( t ) has some sectional curvaturewhich is larger than κ/
2, and we can thus apply the observation in (4.1) toconclude that the flow is type IIb. (cid:3)
Remark 4.1.
The argument above shows the following general conclusionabout bisectional curvature. If (
X, ω ) is a compact K¨ahler manifold con-taining a rational curve C , thensup X Bisec ω > π R C ω . In many special cases we know the existence of rational curves, for example K A ∈ H ( X, Z ), if the Gromov-Witteninvariant GW XA,ω, of genus 0 is defined, for instance when X is a Calabi-Yau3-fold, and does not vanish (cf. [16]), then there is a rational curve C ⊂ X with R C ω R A ω by the following argument. Let J k be a sequence of regu-lar almost-complex structures compatible with ω and which converge to thecomplex structure J of X . The assumption that GW XA,ω, = 0 implies thatfor every k there is a J k -holomorphic curve f k : CP → X representing A .If f k converges to a limit when k → ∞ , then we obtain a rational curve in X , and if not, we still have one by the bubbling process (see [16]). Becauseof the importance of Ricci-flat K¨ahler metrics on Calabi-Yau 3-folds, this lower bound of the supremum of the bisectional curvatures may have someindependent interest. Proof of Theorem 1.5.
Assume κ ( X ) = 0.Since K X is semiample and κ ( X ) = 0, we conclude that K ℓX is trivial forsome ℓ >
1, i.e. X is Calabi-Yau.Case 1: X is not a finite quotient of a torus, which is equivalent to thefact that any Ricci-flat K¨ahler metric constructed by Yau [30] is not flat.Let ω ∞ be the unique Ricci-flat K¨ahler metric in the class [ ω X ], and let x ∈ X be a point with a 2-plane π ⊂ T x X with Sec ω ∞ ( π ) > κ > κ >
0. Indeed we may choose x to be any point where ω ∞ is not flat, and since the Ricci curvature vanishes, there must be a positivesectional curvature at x . Thanks to [3], we know that the solution ω ( t ) of theunnormalized flow (1.5) converges smoothly to ω ∞ as t → ∞ . In particularSec ω ( t ) ( π ) > κ/ > t large. Thanks to the observation in (4.1), theflow is type IIb.Case 2: X is a finite quotient of a torus. In this case the unnormalizedflow (1.5) converges smoothly to a flat metric ω ∞ [3]. In fact, it is easy to seethat this convergence is exponentially fast (cf. [18]). Briefly, one considersthe Mabuchi energy Y ( t ) = Z X |∇ ˙ ϕ | g ( t ) ω ( t ) n , where ω ( t ) = ω X + √− ∂∂ϕ ( t ), and using the smooth convergence of ω ( t )to ω ∞ one easily shows that dY /dt − ηY for some η > t > Y Ce − ηt . Similarly, for k > Y k ( t ) = Z X |∇ k R ˙ ϕ | g ( t ) ω ( t ) n , and using interpolation inequalities and induction on k one proves easilythat dY k /dt − ηY k + C k e − ηt , hence Y k C k e − ηt . Using the Poincar´eand Sobolev-Morrey inequalities, one deduces that k ˙ ϕ k C k ( X,ω X ) C k e − ηt ,and the exponential smooth convergence of ω ( t ) to ω ∞ follows immediately.Therefore there are constant C, η > | Rm( ω ( t )) | ω ( t ) Ce − ηt , andit follows that the flow is type III. Assume κ ( X ) = n .Case 1: K X is ample. In this case [3] showed that the normalized K¨ahler-Ricci flow (1.2) converges smoothly to the unique K¨ahler-Einstein metric ω ∞ with Ric( ω ∞ ) = − ω ∞ . In particular, the sectional curvatures along thenormalized flow remain uniformly bounded for all positive time. Translatingthis back to the unnormalized flow, we see that the flow is type III.Case 2: K X is not ample. By assumption we have that K X is big andsemiample (so in particular nef). It follows that X is Moishezon and K¨ahler,hence projective. Take ℓ sufficiently large and divisible so that sections of K ℓX give a holomorphic map f : X → CP N with image a normal projectivevariety with at worst canonical singularities and ample canonical divisor.Since K X is not ample, f is not an isomorphism with its image, and by NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 17
Zariski’s main theorem there is a fiber F ⊂ X of f which is positive dimen-sional. Each irreducible component of F is uniruled by a result of Kawamata[12, Theorem 2], and if C is a rational curve contained in F then f ( C ) isa point and so R C c ( X ) = 0. The criterion in Proposition 1.4 then showsthat the flow is type IIb. Assume < κ ( X ) < n .Case 1: the generic fiber X y of the Iitaka fibration π : X → B is not afinite quotient of a torus. Let y ∈ B \ S ′ and fix a point x ∈ X y = π − ( y ).If ω ( t ) is the solution of the normalized K¨ahler-Ricci flow (1.2), and ˜ ω ( s ) isthe solution of the unnormalized flow (1.5), then we have that˜ ω ( s ) = e t ω ( t ) , s = e t − . We have proved in Theorem 1.3 that if U ⊃ X y is the preimage of a suf-ficiently small neighborhood of y , then there is a sequence t k → ∞ suchthat ( U, e t k ω ( t k ) , x ) converge smoothly in the sense of Cheeger-Gromov to( X y × C m , ω ∞ , ( x, ω ∞ is the product of a Ricci-flat K¨ahler metricon X y and a flat metric on C m . Since X y is not a finite quotient of a torus,we conclude that ω ∞ is not flat, and so there is some point x ′ ∈ X y anda 2-plane π ⊂ T x ′ ( X y × { } ) with Sec ω ∞ ( π ) > κ > κ >
0. Because of the smooth Cheeger-Gromov convergence, we concludethat (up to renaming the sequence t k ) there are 2-planes π k ⊂ T x ′ X withSec ˜ ω ( t k ) ( π k ) > κ/ . By the observation in (4.1), the flow is type IIb.Case 2: the generic fiber X y is a finite quotient of a torus, and S = ∅ .We assume first that the Iitaka fibration π : X → B is a smooth submersionwith fibers complex tori (not just finite quotients of tori). In this case, if X is furthermore projective and [ ω X ] is rational, Fong-Zhang [5] adapted theestimates of Gross-Tosatti-Zhang [8] to prove that the solution ω ( t ) of thenormalized K¨ahler-Ricci flow (1.2) has uniformly bounded curvature for alltime. The projectivity and rationality assumptions were recently removedin [11]. Translating this back to the unnormalized flow, we see that the flowis type III.Next, we treat the general case when the fibers are finite quotients of tori,and still S = ∅ . Since π is a proper submersion, it is a smooth fiber bundle sogiven any y ∈ B we can find a small coordinate ball U ∋ y such that thereis a diffeomorphism f : U × F → π − ( U ) compatible with the projectionsto U , with F diffeomorphic to a finite quotient of a torus. We pull back thecomplex structure from π − ( U ), so we obtain a (in general non-product)complex structure on U × F which makes the map f biholomorphic. Let˜ F → F be a finite unramified covering with ˜ F diffeomorphic to a torus,and put the pullback complex structure on U × ˜ F (again, in general not theproduct complex structure) so that the projection p : U × ˜ F → U × F isholomorphic. The projection π U : U × ˜ F → U equals π ◦ f ◦ p , and so isholomorphic, and therefore every fiber π − U ( y ) , y ∈ U is a compact K¨ahlermanifold diffeomorphic to a torus, hence it is biholomorphic to a torus (see e.g. [4, Proposition 2.9]). Therefore f ◦ p : U × ˜ F → π − ( U ) is a holomor-phic finite unramified covering, compatible with the projections to U , and π U : U × ˜ F → U is a holomorphic submersion with fibers biholomorphicto complex tori, and with total space K¨ahler. We can then use the localestimates in [11, 5, 8] to conclude that the pullback of the normalized flowto U × ˜ F has bounded curvature. But the metrics along this flow are allinvariant under the group of holomorphic deck transformations of f ◦ p , andtherefore descend to the flow on π − ( U ), which has also bounded curvature.Since y was arbitrary, we conclude that the flow on X is of type III. (cid:3) Lastly, we give the proof of Theorem 1.6.
Proof of Theorem 1.6.
Let X be a minimal K¨ahler surface with κ ( X ) = 1and π : X → B an elliptic fibration which is not a submersion everywhere.Thanks to Kodaira’s classification of the singular fibers of elliptic surfaces[2, V.7], we see that either some singular fiber of π contains a rational curve C or otherwise the only singular fibers are of type mI , m >
1, i.e. smoothelliptic curves with nontrivial multiplicity.In the first case Kodaira’s canonical bundle formula [2] gives(4.4) K X = π ∗ ( K B ⊗ L ) ⊗ O X i ( m i − F i ! , for some line bundle L on B , where m i is the multiplicity of the component F i , i.e. π ∗ ( P i ) = m i F i , as Weil divisors, where S ′ = { P i } ⊂ B are all thecritical values of π . Then if ℓ is sufficiently large so that ℓ (1 − /m i ) ∈ N for all i , then we have K ℓX = π ∗ ( K B ⊗ L ) ℓ ⊗ O X i ℓ ( m i − m i P i !! , and since π ( C ) is a point, this shows that R C c ( X ) = 0, and so we concludethat the flow is type IIb by Proposition 1.4,It remains to show that if the only singular fibers are multiples of a smoothelliptic curve, then the flow is of type III, i.e. that along the normalized flow(1.2) the curvature remains uniformly bounded for all time.On compact sets away from the singular fibers, this is true thanks to [5](cf. [8, 11, 7]). Let then ∆ ⊂ B be the unit disc in some coordinate chartsuch that S ′ ∩ ∆ = { y } , the center of the disc, so the fiber X y is of type mI for some m > y ), and let U = π − (∆).Then, thanks to the local description of such singular fibers [6, Proposition1.6.2], we have a commutative diagram(4.5) ˜ U p −−−−→ X ˜ π y y π ∆ q −−−−→ ∆ NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 19 where q ( z ) = z m , the map ˜ π : ˜ U → ∆ is a holomorphic submersion withfibers elliptic curves, and the map p : ˜ U → U is a holomorphic finite unram-ified covering. If we can show that the pullback of the normalized flow to ˜ U has bounded curvature, then the same is true for the flow on U , and since y ∈ S ′ was arbitrary, we would conclude that the flow on X is of type III.As before let χ be the restriction of ℓ ω F S to ∆ ⊂ B ⊂ P H ( K ℓX ). This isa smooth semipositive form on ∆, although χ may not be positive definite atthe center of the disc. On ∆ let v = | z | /m − ψ , where ψ is a potential for χ on ∆ and define ω B = χ + √− ∂∂v , which is an orbifold flat K¨ahler metricon ∆, so that q ∗ ω B is the Euclidean metric on ∆ (and so q ∗ v is smooth). Forsimplicity we will also denote π ∗ v by v and π ∗ ω B by ω B , so that with thisnotation we have that p ∗ v is smooth on ˜ U . Up to shrinking ∆, we may alsoassume that v is defined in a neighborhood of ∆. Thanks to the estimate(2.1) for the normalized flow, which was proved in [5] (see also [21]), andsince on ∂U the semipositive form χ is uniformly equivalent to π ∗ ω B , weconclude that there is a constant C such that on ∂U we have C − ( π ∗ ω B + e − t ω X ) ω ( t ) C ( π ∗ ω B + e − t ω X ) , for all t >
0. Therefore on ∂ ˜ U we get(4.6) C − (˜ π ∗ q ∗ ω B + e − t p ∗ ω X ) p ∗ ω ( t ) C (˜ π ∗ q ∗ ω B + e − t p ∗ ω X ) . Our goal is to prove the same estimate on the whole of ˜ U . We follow thesame strategy as in [5] (cf. [25, 21]). Recall [22] that the K¨ahler-Ricci flowis of the form ω ( t ) = (1 − e − t ) χ + e − t ω X + √− ∂∂ϕ ( t ) , and the potentials ϕ satisfy a uniform L ∞ bound | ϕ ( t ) | C for all t > ϕ ( t ) = ϕ ( t ) − (1 − e − t ) v , then ˜ ϕ is still uniformly bounded and ω ( t ) = (1 − e − t ) ω B + e − t ω X + √− ∂∂ ˜ ϕ ( t ) holds on U \ S . The function˜ ϕ may not be smooth on U ∩ S , but after pulling back to ˜ U it becomes p ∗ ˜ ϕ = p ∗ ϕ − (1 − e − t )˜ π ∗ q ∗ v which is smooth everywhere, and we have p ∗ ω ( t ) = (1 − e − t )˜ π ∗ q ∗ ω B + e − t p ∗ ω X + √− ∂∂p ∗ ˜ ϕ ( t ) . The parabolic Schwarz Lemma calculation (cf. [21, 31]) applied to the map˜ π : ( ˜ U , p ∗ ω ( t )) → (∆ , q ∗ ω B ) gives on ˜ U (cid:18) ∂∂t − ∆ p ∗ ω ( t ) (cid:19) (tr p ∗ ω ( t ) (˜ π ∗ q ∗ ω B ) − p ∗ ˜ ϕ ( t )) − tr p ∗ ω ( t ) (˜ π ∗ q ∗ ω B ) + 4 , for t large. Since the quantity tr p ∗ ω ( t ) (˜ π ∗ q ∗ ω B ) − p ∗ ˜ ϕ ( t ) is uniformly boundedon ∂ ˜ U thanks to (4.6) and the bound on ˜ ϕ , the maximum principle gives(4.7) tr p ∗ ω ( t ) (˜ π ∗ q ∗ ω B ) C, on ˜ U × [0 , ∞ ) . Let now y be any point in ∆ and consider the fiber ˜ X y =˜ π − ( y ), which is a smooth elliptic curve. Restricting to ˜ X y and using ˜ π ∗ q ∗ ω B Cp ∗ ω ( t ), as in [21, Corollary 5.2] or [25, (3.9)], we easily seethat(4.8) osc ˜ X y ( e t p ∗ ˜ ϕ ) C, independent of y ∈ ∆ and t >
0. We then let ˆ ϕ y ( t ) to be the averageof p ∗ ˜ ϕ ( t ) on ˜ X y with respect to the volume form p ∗ ω X | ˜ X y . This definesa smooth function on ∆, uniformly bounded for all t >
0, and we denoteits pullback to ˜ U by ˆ ϕ ( t ). The bound (4.8) gives sup ˜ U | e t ( p ∗ ˜ ϕ − ˆ ϕ ) | C independent of t >
0. Then a calculation as in [5] (see also [21, 25]), gives (cid:18) ∂∂t − ∆ p ∗ ω ( t ) (cid:19) (log tr p ∗ ω ( t ) ( e − t p ∗ ω X ) − Ae t ( p ∗ ˜ ϕ − ˆ ϕ )) − tr p ∗ ω ( t ) ( p ∗ ω X )+ CAe t , if A is large enough. Since the quantity log tr p ∗ ω ( t ) ( e − t p ∗ ω X ) − Ae t ( p ∗ ˜ ϕ − ˆ ϕ )is uniformly bounded on ∂ ˜ U thanks to (4.6) and the bound on e t ( p ∗ ˜ ϕ − ˆ ϕ ),the maximum principle gives tr p ∗ ω ( t ) ( e − t p ∗ ω X ) C on ˜ U × [0 , ∞ ). Addingthis estimate to (4.7) we conclude that(4.9) ˜ π ∗ q ∗ ω B + e − t p ∗ ω X Cp ∗ ω ( t ) , on ˜ U × [0 , ∞ ) . Now the main theorem of [23] shows that | ˙ ϕ | C holds on X × [0 , ∞ ), where ˙ ϕ = ∂ϕ/∂t . We write the flow on X as the paraboliccomplex Monge-Amp`ere equation ∂∂t ϕ = log e t ((1 − e − t ) χ + e − t ω X + √− ∂∂ϕ ) Ω − ϕ, where Ω is a smooth volume form on X with √− ∂∂ log Ω = χ . Pullingback to ˜ U , we may write it as ∂∂t p ∗ ˜ ϕ = log e t ((1 − e − t )˜ π ∗ q ∗ ω B + e − t p ∗ ω X + √− ∂∂p ∗ ˜ ϕ ) p ∗ Ω − p ∗ ˜ ϕ, and still have | ∂p ∗ ˜ ϕ/∂t | C (recall that p ∗ ˜ ϕ is smooth on ˜ U ). Then wesee that on ˜ U the metrics p ∗ ω ( t ) and ˜ π ∗ q ∗ ω B + e − t p ∗ ω X have uniformlyequivalent volume forms, both comparable to e − t p ∗ Ω, and this togetherwith (4.9) finally proves that (4.6) holds on all of ˜ U × [0 , ∞ ).Now that we have established (4.6), we can use the same procedure asin [11, 5, 8] to conclude that the pullback of the normalized flow to ˜ U hasbounded curvature, and we are done. (cid:3) For example, if X is a minimal properly elliptic K¨ahler surface such thatthe sections of K X already give rise to the Iitaka fibration π : X → B , then itis easy to see using Kodaira’s canonical bundle formula (4.4) that there canbe no multiple fibers of π , so in this case if S = ∅ then the flow is always oftype IIb. It is also easy to construct examples of minimal properly ellipticK¨ahler surfaces with S = ∅ and all singular fibers of type mI , in whichcase the flow is of type III. For example we can take a compact Riemannsurface Σ of genus g > f : Σ → Σ of order m withfinitely many fixed points, then take the elliptic curve E = C / ( Z ⊕ i Z ), and NFINITE TIME SINGULARITIES OF THE K ¨AHLER-RICCI FLOW 21 consider the free Z m -action on Σ × E generated by ( x, z ) ( f ( x ) , z + 1 /m ), x ∈ Σ , z ∈ E . Then π : X = (Σ × E ) / Z m → B = Σ / Z m is a minimalproperly elliptic K¨ahler surface with singular fibers of type mI above thefixed points of f , and π is the Iitaka fibration of X . Proof of Corollary 1.7.
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Department of Mathematics, Northwestern University, 2033 Sheridan Road,Evanston, IL 60208
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