The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
aa r X i v : . [ m a t h . DG ] J a n THE K ¨AHLER-RICCI FLOW, RICCI-FLAT METRICS ANDCOLLAPSING LIMITS
VALENTINO TOSATTI, BEN WEINKOVE, AND XIAOKUI YANG
Abstract.
We investigate the K¨ahler-Ricci flow on holomorphic fiberspaces whose generic fiber is a Calabi-Yau manifold. We establish uni-form metric convergence to a metric on the base, away from the singu-lar fibers, and show that the rescaled metrics on the fibers converge toRicci-flat K¨ahler metrics. This strengthens previous work of Song-Tianand others. We obtain analogous results for degenerations of Ricci-flatK¨ahler metrics. Introduction
Background.
This paper establishes convergence results for collapsingmetrics along the K¨ahler-Ricci flow and along families of Ricci-flat K¨ahlermetrics. We give now some background and motivation for these results,glossing over some technicalities for the moment.Let X be a compact K¨ahler manifold, and suppose we have a holomorphicsurjective map π : X → B onto another compact K¨ahler manifold B , with0 < dim B < dim X. We consider a family of K¨ahler metrics ( ω ( t )) t ∈ [0 , ∞ ) on X with the property that the cohomology classes [ ω ( t )] ∈ H , ( X, R ) havethe limiting behavior: [ ω ( t )] → π ∗ α, in H , ( X, R ) , where α is a K¨ahler class on B . Moreover, we suppose that the K¨ahlermetrics ω ( t ) solve one of the two natural PDEs: the K¨ahler-Ricci flow orthe Calabi-Yau equation. Note that from topological considerations, themetrics ω ( t ) are automatically volume collapsing , meaning that Z X ω ( t ) dim X = [ ω ( t )] dim X → , as t → ∞ , since B has dimension strictly less than dim X .A natural question is: does ( X, ω ( t )) converge to ( B, ω B ), where ω B is aunique canonical metric in the class α ?In the cases we consider, the existence of the “canonical metric” hasalready been established, and the remaining point is to establish convergencein the strongest possible sense. There are weak notions of convergence, suchas convergence of ω ( t ) to ω B as currents, or convergence at the level of K¨ahler Research supported in part by NSF grants DMS-1308988, DMS-1332196 and DMS-1406164. The first-named author is supported in part by a Sloan Research Fellowship. potentials in C ,α , say, for α ∈ (0 , ω ( t ) to ω B in the C norm. This kind of convergence impliesglobal Gromov-Hausdorff convergence. The main point of this paper is thatwe can obtain, in many cases, convergence of the stronger type when beforeonly weaker convergence was known.To be more explicit, we discuss now the simplest case of such collapsingfor the K¨ahler-Ricci flow, which is already completely understood. Let X bea product X = E × B where B is a compact Riemann surface of genus largerthan 1 and E is a one-dimensional torus (an elliptic curve). We consider theK¨ahler-Ricci flow [21, 6](1.1) ∂∂t ω = − Ric( ω ) − ω, ω | t =0 = ω , starting at an arbitrary initial K¨ahler metric ω on X . A solution to (1.1)exists for all time [51]. Note that (1.1) is really the normalized K¨ahler-Ricci flow. It has the property that as t → ∞ the K¨ahler classes [ ω ( t )] ∈ H , ( X, R ) converge to the class π ∗ α where α = [ ω B ], for ω B the uniqueK¨ahler-Einstein metric on B satisfying(1.2) Ric( ω B ) = − ω B . Here π : X → B is the projection map.This example was first studied by Song-Tian [39] who considered themuch more general situation of a holomorphic fiber space π : X → B ofKodaira dimension one, whose generic fibers are elliptic curves. For theproduct X = E × B , the results of [39, 45, 15, 12] show that one obtains C ∞ convergence of ω ( t ) to ω B . Moreover, the curvature of ω ( t ) and its covariantderivatives remain bounded as t → ∞ . Geometrically, the metrics shrinkin the directions of E and we obtain convergence to the K¨ahler-Einsteinmetric ω B on B . Moreover, if we look at the evolving metrics restricted toany given fiber E and rescale appropriately, they converge to a flat metricon E .A natural generalization of this result is to consider X = E × B wherenow E is a Calabi-Yau manifold (i.e. c ( E ) = 0 in H , ( E, R )) and B is acompact K¨ahler manifold with c ( B ) <
0. The behavior of the K¨ahler-Ricciflow on the manifolds E and B separately is well-known by the results of Cao[6]: on E the unnormalized K¨ahler-Ricci flow ∂∂t ω = − Ric( ω ) converges toa K¨ahler Ricci-flat metric, and on B the normalized K¨ahler-Ricci flow (1.1)converges to the unique K¨ahler-Einstein metric satisfying (1.2). However,the behavior of the flow on the product X = E × B is in general harder tounderstand than the two-dimensional example above. A source of difficultyis that, except in dimension one, Calabi-Yau manifolds may not admit flatmetrics.In the case when E does admit a flat K¨ahler metric, it was shown by Gill[15], Fong-Zhang [12] that the same results hold as in the case of a productof Riemann surfaces. In particular, the curvature remains bounded alongthe flow. These methods fail for a general E . Indeed, by starting the flow OLLAPSING LIMITS 3 at a product of a non-flat Ricci-flat metric on E with a K¨ahler-Einsteinmetric on B , it is easy to check that the curvature along the K¨ahler-Ricciflow (1.1) must blow up. One can still obtain “weak” convergence of themetrics [40, 12]. Indeed, the evolving metrics along the flow are comparableto the “model” collapsing metrics, which implies in particular that the fibersshrink and one obtains convergence to ω B at the level of potentials in the C ,α norm. However, until now the question of whether the metrics ω ( t )converge (in C say) to ω B has remained open.Our results show that in fact the metrics ω ( t ) converge exponentially fastin the C norm to ω B , which implies in particular Gromov-Hausdorff con-vergence of ( X, ω ( t )) to ( B, ω B ). Moreover, we show that if one restricts toa fiber E over y ∈ B then the metrics ω ( t ), appropriately rescaled, convergeexponentially fast to a Ricci-flat K¨ahler metric on E and this convergenceis uniform in y . All of these statements are new in this setting, and requirenew estimates, which we explain later.Of course, the assumption that X is a product is rather restrictive. In fact,the same results hold when π : X → B is a holomorphic submersion, exceptthat the K¨ahler-Einstein equation (1.2) is replaced by the twisted K¨ahler-Einstein equation [39, 40, 11]. But even this is not sufficiently general, sincewe also need to allow π : X → B to admit singular fibers, and for B itselfto be possibly singular. More precisely, we assume that π : X → B isa fiber space, namely a surjective holomorphic map with connected fibers,and B is an irreducible normal projective variety. In this case we obtain thesame metric convergence results on compact subsets away from the singularfibers (but this time not exponential). Again it was only known before thatconvergence occurs at the level of potentials [40].The latter situation is really quite general. Indeed, if one believes theAbundance Conjecture from algebraic geometry (that on a compact K¨ahlermanifold the canonical bundle K X is nef implies K X is semiample) thenwhenever the K¨ahler-Ricci flow has a long-time solution, there exists sucha fiber space map π : X → B . For the above discussion to apply, we alsoneed to remove the two extreme cases: when B has dimension 0 or dim X .In the first case X is Calabi-Yau and it is known by the work of Cao [6] thatthe K¨ahler-Ricci flow (unnormalized) converges smoothly to Yau’s Ricci-flatK¨ahler metric [62]. In the second case X is of general type (in particular, X is projective), and the normalized K¨ahler-Ricci flow converges to a K¨ahler-Einstein current on X [61, 51, 65], smoothly away from a subvariety.On a general projective variety X , the K¨ahler-Ricci flow may encounterfinite time singularities. According to the analytic minimal model program[39, 50, 40, 42] the K¨ahler-Ricci flow will perform a surgery at each finitetime singularity, corresponding to a birational operation of the minimalmodel program from algebraic geometry, and arrive at the minimal model X min with nef canonical bundle (see also [5, 9, 10, 12, 26, 36, 38, 41, 43,44, 46, 47, 51, 61]). It is then expected that the flow on X min will convergeat infinity to a canonical metric on the lower dimensional canonical model V. TOSATTI, B. WEINKOVE, AND X. YANG X can . Our results here are concerned with a special case of this last step,where X min = X is smooth with semiample canonical bundle and X can iswhat we call B .Finally, we discuss the Ricci-flat K¨ahler metrics. A similar situation ariseshere, where one considers a Calabi-Yau manifold X which admits a holo-morphic fiber space structure π : X → B as above (possibly with singularfibers and with B singular), and studies the collapsing behavior of the Ricci-flat K¨ahler metrics ω ( t ) in the class [ π ∗ χ ] + e − t [ ω X ] (where ω X is a K¨ahlermetric on X and χ on B ), as t tends to infinity. Again one expects to havecollapsing to a canonical K¨ahler metric ω B on B \ S ′ (where S ′ denotes thesingularities of B together with the images of the singular fibers). Further-more, one would like to understand if a Gromov-Hausdorff limit ( X ∞ , d ∞ )of ( X, ω ( t )) exists (without passing to subsequences), whether it is suffi-ciently regular, and how it is related to ( B \ S ′ , ω B ). It is expected [55, 56]that ( X ∞ , d ∞ ) contains an open and dense subset which is a smooth Rie-mannian manifold, that ( X ∞ , d ∞ ) is isometric to the metric completion of( B \ S ′ , ω B ), and that X ∞ is homeomorphic to B . By comparison, in thecase when the volume of the Ricci-flat metrics is non-collapsing, we have avery thorough understanding of their limiting behavior, thanks to the resultsin [8, 31, 32, 33, 37, 53], and all the analogous questions have affirmativeanswers.This setup is also related to the Strominger-Yau-Zaslow picture of mir-ror symmetry [48]. Indeed, Kontsevich-Soibelman [25], Gross-Wilson [20]and Todorov (see [28, p. 66]) formulated a conjecture about certain col-lapsed limits of Ricci-flat K¨ahler metrics on Calabi-Yau manifolds, wherethe K¨ahler class is fixed but the complex structure degenerates (to a so-called large complex structure limit ). This might seem very different fromwhat we are considering here, where the complex structure is fixed and theK¨ahler class degenerates, but in the case when the Calabi-Yau manifolds areactually hyperk¨ahler, one may perform a hyperk¨ahler rotation and often re-duce this conjecture to studying the collapsed limits of Ricci-flat metrics inthe precise setup of Theorem 1.3 (see [18]), although in this case the smoothfibers are always tori. This approach was pioneered by Gross-Wilson [20],who solved this conjecture for certain elliptically fibered K ω ( t ) converge in the C norm to π ∗ ω B on compact sets away fromthe singular fibers, improving earlier results of the first-named author [54]who proved weak convergence as currents and in the C ,α norm of potentials.Also, the rescaled metrics along the fibers converge to Ricci-flat metrics. Fur-thermore, we are able to show in Corollary 1.4 that any Gromov-Hausdorfflimit contains an open and dense subset which is a smooth Riemannianmanifold (this fact in itself is quite interesting and does not follow from thegeneral structure theory of Cheeger-Colding [7]). These new results solvesome questions in [55, 56]. OLLAPSING LIMITS 5
Metric collapsing along K¨ahler-Ricci flow.
Let (
X, ω ) be a com-pact K¨ahler manifold with canonical bundle K X semiample. By definitionthis means that there exists ℓ > K ℓX is glob-ally generated. If we choose ℓ sufficiently large and divisible, the sections in H ( X, K ℓX ) give a holomorphic map to projective space, with image B an ir-reducible normal projective variety, and we obtain a surjective holomorphicmap π : X → B with connected fibers (see e.g. [27, Theorem 2.1.27]). Thedimension of B is the Kodaira dimension of X , written κ ( X ). We assume0 < κ ( X ) < dim X .We let m = κ ( X ) = dim B and n + m = dim X . The generic fiber X y = π − ( y ) of π has K ℓX y holomorphically trivial, so it is a Calabi-Yaumanifold of dimension n .We first consider a special case, where some technicalities disappear andfor which our results are stronger. Assume:( ∗ ) π : X → B is a holomorphic submersion ontoa compact K¨ahler manifold B with c ( B ) < . As we will see in Section 2, this assumption implies that the canonical bundleof X is semiample. In this case, all the fibers X y = π − ( y ) are Calabi-Yaumanifolds of dimension n (there are no singular fibers). Song-Tian [40] (seealso [11]) showed that B admits a unique smooth twisted K¨ahler-Einsteinmetric ω B solving(1.3) Ric( ω B ) = − ω B + ω WP , where ω WP is the Weil-Petersson form on B (see Section 2.1 below). Wewill often write ω B for π ∗ ω B .By Yau’s theorem [62], each of the fibers X y of π : X → B admits aunique Ricci-flat K¨ahler metric cohomologous to ω | X y . Write ω SRF ,y forthis metric on X y . To explain this notation: later we will define a semiRicci-flat form ω SRF on X which restricts to ω SRF ,y on X y .Our first result concerns the normalized K¨ahler-Ricci flow(1.4) ∂∂t ω = − Ric( ω ) − ω, ω (0) = ω , on X , starting at ω . We show that in this case we obtain exponentially fastmetric collapsing of ( X, ω ( t )) to ( B, ω B ) along the K¨ahler-Ricci flow. Theorem 1.1.
Let ω = ω ( t ) be the solution of the K¨ahler-Ricci flow (1.4)on X satisfying assumption ( ∗ ) . Then the following hold: (i) As t → ∞ , ω converges exponentially fast to ω B . Namely, there existpositive constants C, η > such that (1.5) k ω − ω B k C ( X,ω ) Ce − ηt , for all t > . (ii) The rescaled metrics e t ω | X y converge exponentially fast to ω SRF ,y oneach fiber X y , uniformly in y . Namely, there exist positive constants V. TOSATTI, B. WEINKOVE, AND X. YANG
C, η > such that (1.6) k e t ω | X y − ω SRF ,y k C ( X y ,ω | Xy ) Ce − ηt , for all t > , y ∈ B. Moreover, if we fix α ∈ (0 , then for each y ∈ B , e t ω | X y convergesto ω SRF ,y in C α ( X y , ω | X y ) . (iii) As t → ∞ , ( X, ω ) converges in the Gromov-Hausdorff sense to ( B, ω B ) . Next, we consider the general case of (
X, ω ) with canonical bundle K X semiample and 0 < κ ( X ) < dim X . As above, we have a surjective holo-morphic map π : X → B with connected fibers (i.e. a fiber space), but now B is only an irreducible normal projective variety (possibly singular). Inaddition, π may have critical values. If S ′ ⊂ B denotes the singular set of B together with the set of critical values of π , and we define S = π − ( S ′ ), then S ′ is a proper analytic subvariety of B , S is a proper analytic subvarietyof X , and π : X \ S → B \ S ′ is a submersion between smooth manifolds.Therefore all fibers X y = π − ( y ) with y ∈ B \ S ′ are smooth n -folds (alldiffeomorphic to each other) with torsion canonical bundle.As shown in [40, Theorem 3.1], there is a smooth K¨ahler metric ω B on B \ S ′ satisfying the twisted K¨ahler-Einstein equationRic( ω B ) = − ω B + ω WP , on B \ S ′ , for ω WP the (smooth) Weil-Petersson form on B \ S ′ . For y ∈ B \ S ′ , let ω SRF ,y be the unique Ricci-flat K¨ahler metric on the fiber X y cohomologousto ω | X y .Our result is that, in this more general setting, we obtain metric collapsingof the normalized K¨ahler-Ricci flow ω ( t ) to ω B on compact subsets awayfrom the singular set S . Theorem 1.2.
Let ω = ω ( t ) be the solution of the K¨ahler-Ricci flow (1.4)on X . Then the following hold: (i) For each compact subset K ⊂ X \ S , (1.7) k ω − ω B k C ( K,ω ) → , as t → ∞ . (ii) The rescaled metrics e t ω | X y converge to ω SRF ,y on each fiber X y uni-formly in y for compact subsets of B \ S ′ . Namely, for each compactset K ′ ⊂ B \ S ′ (1.8) sup y ∈ K ′ k e t ω | X y − ω SRF ,y k C ( X y ,ω | Xy ) → , as t → ∞ . Moreover, if we fix α ∈ (0 , then for each y ∈ B \ S ′ , e t ω | X y converges to ω SRF ,y in C α ( X y , ω | X y ) . An interesting, but still open, question is whether the convergence resultsof Theorems 1.1 and 1.2 hold with the C norms replaced by the C ∞ norms.This was conjectured by Song-Tian [39] in the case of elliptic surfaces. More-over, one expects the convergence to be always exponentially fast. Smooth OLLAPSING LIMITS 7 convergence has indeed been proved by Fong-Zhang [12] in the case whenthe smooth fibers are tori, adapting the techniques of Gross-Tosatti-Zhang[18] for Calabi-Yau metrics.1.3.
Collapsing of Ricci-flat metrics.
Let X be a compact K¨ahler ( n + m )-manifold with c ( X ) = 0 in H ( X, R ) (i.e. a Calabi-Yau manifold),and let ω X be a Ricci-flat K¨ahler metric on X . Suppose that we have aholomorphic map π : X → Z with connected fibers, where ( Z, ω Z ) is acompact K¨ahler manifold, with image B = π ( X ) ⊂ Z an irreducible normalsubvariety of Z of dimension m >
0. Then the induced surjective map π : X → B is a fiber space, and if S ′ ⊂ B denotes the singular set of B together with the set of critical values of π , and S = π − ( S ′ ), then S ′ isa proper analytic subvariety of B , S is a proper analytic subvariety of X ,and π : X \ S → B \ S ′ is a submersion between smooth manifolds. This isthe same setup as in [54, 18, 19, 22]. The manifold Z here plays only anauxiliary role, and in practice there are two natural choices for Z , namely Z = B if B is smooth, or Z = CP N if B is a projective variety (like in thesetup of the K¨ahler-Ricci flow).The fibers X y = π − ( y ) with y ∈ B \ S ′ are smooth Calabi-Yau n -folds (alldiffeomorphic to each other). As in the previous section, we write ω SRF ,y for the unique Ricci-flat K¨ahler metric on X y cohomologous to ω X | X y , for y ∈ B \ S ′ .Write χ = π ∗ ω Z , which is a smooth nonnegative (1 ,
1) form on X . Thenthe cohomology class α t = [ χ ] + e − t [ ω X ] , t < ∞ , is K¨ahler. By Yau’s Theorem [62] there exists a unique Ricci-flat K¨ahlermetric ω = ω ( t ) ∈ α t on X . On the other hand, the class [ χ ] is not K¨ahlerand hence the family of metrics ω ( t ) must degenerate as t → ∞ .Note that in [54, 18, 19, 22] the class α t was replaced by [ χ ] + t [ ω X ] , Let ω = ω ( t ) ∈ α t be Ricci-flat K¨ahler metrics on X asdescribed above. Then the following hold: (i) For each compact set K ⊂ X \ S , (1.9) k ω − ω B k C ( K,ω ) → , as t → ∞ . V. TOSATTI, B. WEINKOVE, AND X. YANG (ii) The rescaled metrics e t ω | X y converge to ω SRF ,y on each fiber X y uni-formly in y for compact subsets of B \ S ′ . Namely, for each compactset K ′ ⊂ B \ S ′ , (1.10) sup y ∈ K ′ k e t ω | X y − ω SRF ,y k C ( X y ,ω | Xy ) → , as t → ∞ . Furthermore, if we fix α ∈ (0 , then for each y ∈ B \ S ′ , e t ω | X y converges to ω SRF ,y in C α ( X y , ω | X y ) . The statement that e t ω | X y → ω SRF ,y (which can be improved to thesmooth topology [60]) solves [55, Question 4.1] and [56, Question 3]. As inthe case of the K¨ahler-Ricci flow, it remains an interesting open question(cf. [55, Question 4.2], [56, Question 4]) whether the collapsing in (1.9) isin the smooth topology.Theorem 1.3 was first proved by Gross-Wilson [20] in the case when X isa K B is CP and the only singular fibers of π are of type I . Infact they also proved that the collapsing in (1.9) is in the smooth topology.Their proof is of a completely different flavor, constructing explicit almostRicci-flat metrics on X by gluing ω B + e − t ω SRF away from S with a suitablemodel metric near the singularities, and then perturbing this to the honestRicci-flat metrics ω ( t ) (for t large). In higher dimensions when there are nosingular fibers ( S = ∅ ) a similar perturbation result was obtained by Fine[11]. However it seems hopeless to pursue this strategy in higher dimensionswhen there are singular fibers, since in general there is no good local modelmetric near the singularities. Note that such fibrations with S = ∅ are veryspecial, thanks to the results in [59].In [39, Theorem 7.1] Song-Tian took a different approach, and by prov-ing a priori estimates for the Ricci-flat metrics ω ( t ) they showed that theycollapse to ω B in the C ,α topology of K¨ahler potentials (0 < α < X is K B is CP . This was generalized to all dimensionsby the first-named author in [54], who also proved that the fibers shrink inthe C topology of metrics.When X is projective and the smooth fibers X y are tori, Theorem 1.3was proved by Gross-Tosatti-Zhang in [18], together with the improvementof (1.9) to the smooth topology. The projectivity assumption was recentlyremoved in [22]. In this case (and only in this case) the Ricci-flat metrics alsohave locally uniformly bounded sectional curvature on X \ S . The assump-tion that the smooth fibers are tori is used crucially to derive higher-orderestimates for the pullback of ω to the universal cover of π − ( U ), which isbiholomorphic to U × C n , where U is a small ball in B \ S ′ . It is unclear howto obtain similar estimates in our general setup.In the setup of Theorem 1.3 we can also derive some consequences aboutGromov-Hausdorff limits of the Calabi-Yau metrics ω ( t ). In fact, given theuniform convergence proved in (1.9), we can directly apply existing argu-ments in the literature. For example, using the arguments in [18, Theorem1.2] we obtain the following corollary. OLLAPSING LIMITS 9 Corollary 1.4. Assume the same setup as above, and let ( X ∞ , d ∞ ) be ametric space which is the Gromov-Hausdorff limit of ( X, ω ( t i )) for some se-quence t i → ∞ . Then there is an open dense subset X ⊂ X ∞ and a home-omorphism φ : B \ S ′ → X which is a local isometry between ( B \ S ′ , ω B ) and ( X , d ∞ ) . In particular, ( X ∞ , d ∞ ) has an open dense subset which is asmooth Riemannian manifold. Similarly, the arguments in [19, Theorem 1.1] give: Corollary 1.5. Assume furthermore that X is projective and B is a smoothRiemann surface. Then ( X, ω ( t )) have a Gromov-Hausdorff limit ( X ∞ , d ∞ ) as t → ∞ , which is isometric to the metric completion of ( B \ S ′ , ω B ) and X ∞ is homeomorphic to B . Lastly, X ∞ \ X is a finite number of points. This answers questions in [19, 55, 56], in this setup.1.4. Remarks on the proofs. We end the introduction by describingbriefly some key elements in the proofs. For simplicity of the discussion,suppose that X is a product X = E × B with ω E a Ricci-flat metric on E and ω B a K¨ahler-Einstein metric on B , and assume that ω is cohomologousto the product metric. Write ˜ ω = ω B + e − t ω E . It is already known [12] that ω and ˜ ω are uniformly equivalent. Our goal is to show that k ω − ˜ ω k C ( X,ω ) converges to zero. We first show that (cf. Lemma 2.2.(iv))(1.11) ω n + m ˜ ω n + m → , as t → ∞ , exponentially fast. In fact this statement is easily derived from existingarguments in the literature [39, 40, 45, 41, 58].The next step is the estimate (Lemma 2.8)(1.12) tr ω ω B − m Ce − ηt , for uniform positive constants C and η . This uses a maximum principleargument similar to that in the authors’ recent work [58, Proposition 7.3]on the Chern-Ricci flow [14, 57]. Indeed, [58] established in particular expo-nential convergence of solutions to the K¨ahler-Ricci flow on elliptic surfacesand, as we show here, these ideas can be applied here for the case of moregeneral Calabi-Yau fibers.Finally, we have a Calabi-type estimate in the fibers X y for y ∈ B (Lemma2.9) which gives(1.13) k e t ω | X y k C ( X y ,ω | Xy ) C. Then the exponential convergence of ω to ω B , and the other results ofTheorem 1.1 are almost a formal consequence of the three estimates (1.11),(1.12), (1.13). When X is a holomorphic submersion rather than a product,the analogous estimates still hold. When π : X → B has singular fibers then there are a number of technicalcomplications and in particular we are not able to obtain exponential con-vergence (see Lemma 3.1.(iv)). For the case of collapsing Ricci-flat K¨ahlermetrics, there are even more difficulties. A parabolic maximum principleargument has to be replaced by a global argument using the Green’s func-tion and the diameter bound of [53, 64] (see Lemma 4.4). Nevertheless, thebasic outlines of the proofs remain the same.This paper is organized as follows. In Section 2 we study the collapsingof the K¨ahler-Ricci flow on holomorphic submersions, and prove Theorem1.1. In Section 3 we deal with the case with singular fibers and prove The-orem 1.2. The collapsing of Ricci-flat metrics on Calabi-Yau manifolds isdiscussed in Section 4 where we give the proof of Theorem 1.3. Acknowledgments. We are grateful to the referee for useful comments.2. The K¨ahler-Ricci flow on a holomorphic submersion In this section, we give a proof of Theorem 1.1. We assume that X satisfiesassumption ( ∗ ) from the introduction.2.1. Holomorphic submersions. We begin with some preliminary factsabout submersions π : X → B satisfying ( ∗ ) as in the introduction. Thecontent of this section is essentially well-known (see e.g. [39, 40, 54, 12]).We include this brief account for the sake of completeness.Let ( X n + m , ω ) be a compact K¨ahler manifold which admits a submer-sion π : X → B onto a compact K¨ahler manifold B m with c ( B ) < X y = π − ( y ) which are all Calabi-Yau n -manifolds.Thanks to Yau’s theorem [62], there is a smooth function ρ y on X y suchthat ω | X y + √− ∂∂ρ y = ω SRF ,y is the unique Ricci-flat K¨ahler metric on X y cohomologous to ω | X y . If we normalize by R X y ρ y ω n | X y = 0, then ρ y varies smoothly in y and defines a smooth function ρ on X and we let ω SRF = ω + √− ∂∂ρ. This is called a semi Ricci-flat form, because it restricts to a Ricci-flat K¨ahlermetric on each fiber X y . It was first introduced by Greene-Shapere-Vafa-Yau [16] in the context of elliptically fibered K ω SRF need not be positive definite,nevertheless for any K¨ahler metric χ on B we have that ω n SRF ∧ π ∗ χ m is astrictly positive volume form on X . Now recall that the relative pluricanon-ical bundle of π is K ℓX/B = K ℓX ⊗ ( π ∗ K ℓB ) ∗ , where ℓ is any positive integer.Thanks to the projection formula, we have π ∗ ( K ℓX ) = ( π ∗ ( K ℓX/B )) ⊗ K ℓB , and when restricted to any fiber X y , K ℓX/B | X y ∼ = K ℓX y . The fact that X y isCalabi-Yau implies that there is a positive integer ℓ such that K ℓX y is trivial, OLLAPSING LIMITS 11 for all y ∈ B , and from now on we fix such a value of ℓ . Then Grauert’stheorem on direct images [3, Theorem I.8.5] shows that L := π ∗ ( K ℓX/B ) , is a line bundle on B . Since all the fibers of π have trivial K ℓX y , it followsthat K ℓX ∼ = π ∗ π ∗ ( K ℓX ) (see [3, Theorem V.12.1]). We conclude that(2.1) K ℓX = π ∗ ( K ℓB ⊗ L ) . The line bundle L = π ∗ ( K ℓX/B ) is Hermitian semipositive, and the Weil-Petersson form ω WP on B is a semipositive representative of ℓ c ( L ). This isdefined as follows. Choose any local nonvanishing holomorphic section Ψ y of π ∗ ( K ℓX/B ), i.e. a family Ψ y of nonvanishing holomorphic ℓ -pluricanonicalforms on X y , which vary holomorphically in y . The forms Ψ y are definedon the whole fiber X y , but only for y in a small ball in B . Then the Weil-Petersson (1 , ω WP = −√− ∂ y ∂ y log ( √− n Z X y (Ψ y ∧ Ψ y ) ℓ ! , is actually globally defined on B (this is because the fibers have trivial ℓ -pluricanonical bundle, so different choices of Ψ y differ by multiplication bya local holomorphic function in y , and so ω WP is globally defined) and issemipositive definite (this was originally proved by Griffiths [17], and is aby now standard calculation in Hodge theory). One can view ω WP as thepullback of the Weil-Petersson metric from the moduli space of polarizedCalabi-Yau fibers of π , thanks to [13, 49, 52]. From this construction, wesee that ℓω WP is the curvature form of a singular L metric on π ∗ ( K ℓX/B ), so[ ω WP ] = ℓ c ( L ). Notice that here and henceforth, we omit the usual factorof 2 π in the definition of the first Chern class.From (2.1) we see that K ℓX is the pullback of an ample line bundle, hence K X is semiample. It follows also that(2.2) c ( X ) = π ∗ c ( B ) − [ π ∗ ω WP ] . In our setup the class − c ( B ) is K¨ahler, and therefore so is the class − c ( B )+[ ω WP ]. Fix a K¨ahler metric χ in this class. From (2.2) we see that − π ∗ χ iscohomologous to c ( X ), and so there exists a unique smooth positive volumeform Ω on X with √− ∂∂ log(Ω ) = χ, Z X Ω = (cid:18) n + mn (cid:19) Z X ω n ∧ χ m , where from now on we abbreviate π ∗ χ by χ .We now show the existence of a unique twisted K¨ahler-Einstein metric on B . Theorem 2.1. There is a unique K¨ahler metric ω B on B which satisfies (2.3) Ric( ω B ) = − ω B + ω WP . Proof. The proof is identical to [39, Theorem 3.1] or [40, Theorem 3.1], [54,Section 4], so we will just briefly indicate the main points. First, set F = Ω (cid:0) n + mn (cid:1) ω n SRF ∧ χ m , which is a positive function on X . The argument on [54, p.445] shows thatfor every y ∈ B the restriction F | X y is constant, and so F is the pullback ofa function on B . Now solve the complex Monge-Amp`ere equation on B (2.4) ( χ + √− ∂∂v ) m = F e v χ m , where ω B = χ + √− ∂∂v is a K¨ahler metric on B . This equation can besolved thanks to work of Aubin [1] and Yau [62]. We claim that ω B satisfies(2.3) (the converse implication, that every solution of (2.3) must also satisfy(2.4), and therefore must equal ω B , is a simple exercise). Indeed,Ric( ω B ) = Ric( χ ) − √− ∂∂v − √− ∂∂ log F, so it is enough to prove that(2.5) − √− ∂∂ log F = − χ − Ric( χ ) + ω WP . This can be proved by the same calculation as in [54, Proposition 4.1]. (cid:3) Now that we have our twisted K¨ahler-Einstein metric ω B on B , we canuse (2.2) again to see that − π ∗ ω B is cohomologous to c ( X ). Define now asmooth positive volume form(2.6) Ω = (cid:18) n + mn (cid:19) ω n SRF ∧ ω mB . on X . A simple calculation as in [54, Proposition 4.1] shows that √− ∂∂ log(Ω) = ω B , where from now on we abbreviate π ∗ ω B by ω B .2.2. Preliminary estimates for the K¨ahler-Ricci flow. Let now ω = ω ( t ) be a solution of the K¨ahler-Ricci flow(2.7) ∂∂t ω = − Ric( ω ) − ω, ω (0) = ω . We know that a solution exists for all time [51], because K X is semiample(see Section 2.1). Define reference forms ˜ ω = ˜ ω ( t ) by˜ ω = e − t ω SRF + (1 − e − t ) ω B , for ω SRF and ω B as above.There exists a uniform constant T I > t > T I we havethat ˜ ω is K¨ahler. Then the K¨ahler-Ricci flow is equivalent to the paraboliccomplex Monge-Amp`ere equation(2.8) ∂∂t ϕ = log e nt (˜ ω + √− ∂∂ϕ ) n + m Ω − ϕ, ϕ (0) = − ρ, ˜ ω + √− ∂∂ϕ > , OLLAPSING LIMITS 13 where ρ and Ω are defined as above. Indeed, if ϕ solves (2.8) then ω =˜ ω + √− ∂∂ϕ solves (2.7). Conversely, if ω solves (2.7) then we can write ω = ˜ ω + √− ∂∂ϕ with ϕ solving (2.8).We have the following preliminary estimates for the K¨ahler-Ricci flow,most of which are already known by the results in [39, 45, 12, 15]. Lemma 2.2. Let ω = ω ( t ) solve the K¨ahler-Ricci flow as above, and write ϕ = ϕ ( t ) for the solution of (2.8). Then there exists a uniform constant C > such that the following hold. (i) C − ˜ ω ω C ˜ ω , on X × [ T I , ∞ ) , and ω B Cω on X × [0 , ∞ ) . (ii) | ϕ | C (1 + t ) e − t on X × [0 , ∞ ) . (iii) The scalar curvature R of ω satisfies | R | C on X × [0 , ∞ ) . (iv) For any < ε < / there exists C ε such that | ˙ ϕ | C ε e − εt and | ϕ + ˙ ϕ | C ε e − εt on X × [0 , ∞ ) .Proof. For (i), the estimate ω B Cω follows from the Schwarz Lemmaargument of Song-Tian [39, 63]. The bound C − ˜ ω ω C ˜ ω was proved byFong-Zhang [12, Theorem 1.1] (see also [39] for the case of elliptic surfaces).Part (ii) follows from exactly the same argument as in [15, Lemma 4.1],which is a generalization of [45, Lemma 6.7] (see also [58, Lemma 3.4], [12,Theorem 4.1]). Indeed, we claim that for t > T I ,(2.9) (cid:12)(cid:12)(cid:12)(cid:12) e t log e nt ˜ ω n + m Ω (cid:12)(cid:12)(cid:12)(cid:12) C. To see the claim, we have, using (2.6), e t log e nt ˜ ω n + m Ω= e t log e nt ( (cid:0) n + mn (cid:1) e − nt (1 − e − t ) m ω n SRF ∧ ω mB + · · · + e − ( n + m ) t ω n + m SRF ) (cid:0) n + mn (cid:1) ω n SRF ∧ ω mB = e t log(1 + O ( e − t )) , which is bounded.Define now Q = e t ϕ + At , for A a large positive constant to be determined.Then(2.10) ∂Q∂t = e t log (cid:18) e nt (˜ ω + √− ∂∂ϕ ) n + m Ω (cid:19) + A. We wish to bound Q from below. Suppose that ( x , t ) is a point with t > T I at which Q achieves a minimum. At this point we have0 > ∂Q∂t > e t log e nt ˜ ω n + m Ω + A > − C ′ + A, for a uniform C ′ , thanks to (2.9). Choosing A > C ′ gives contradiction.Hence Q is bounded from below and it follows that ϕ > − C (1 + t ) e − t for auniform C . The upper bound for ϕ is similar. Part (iii) is due to Song-Tian [42], who proved a uniform scalar curvaturebound along the K¨ahler-Ricci flow whenever the canonical bundle of X issemiample.Part (iv) follows from the argument of [58, Lemma 6.4] (see also [45,Lemma 6.8]). We include the argument for the sake of completeness. Firstnote that ˙ ϕ satisfies ∂∂t ˙ ϕ = − R − m − ˙ ϕ. Then the bound on scalar curvature (iii) implies that ˙ ϕ is bounded (in factthe bound on ˙ ϕ is proved first in [42] in order to bound scalar curvature).Hence we have a bound | ∂∂t ˙ ϕ | C . Now fix ε ∈ (0 , / 2) and suppose fora contradiction that the estimate ˙ ϕ C ε e − εt fails. Then there exists asequence ( x k , t k ) ∈ X × [0 , ∞ ) with t k → ∞ and ˙ ϕ ( x k , t k ) > ke − εt k . Define γ k = k C e − εt k . Working at the point x k , it follows that ˙ ϕ > k e − εt k on[ t k , t k + γ k ]. Then using the bound (ii), C (1 + t k ) e − t k > Z t k + γ k t k ˙ ϕ dt > γ k k e − εt k = k C e − εt k . Since 2 ε < k → ∞ . The lower bound ˙ ϕ > − C ε e − εt is similar. The bound on | ϕ + ˙ ϕ | follows from this and part (ii). (cid:3) The following lemma is easily derived from Song-Tian [39]. Lemma 2.3. We have (2.11) (cid:18) ∂∂t − ∆ (cid:19) ( ϕ + ˙ ϕ ) = tr ω ω B − m, and (2.12) (cid:18) ∂∂t − ∆ (cid:19) tr ω ω B tr ω ω B + C (tr ω ω B ) C. Proof. The evolution equation (2.11) follows from the two well-known cal-culations: (cid:18) ∂∂t − ∆ (cid:19) ϕ = ˙ ϕ − ( n + m ) + tr ω ˜ ω and (cid:18) ∂∂t − ∆ (cid:19) ˙ ϕ = tr ω ( ω B − ˜ ω ) + n − ˙ ϕ. The first inequality of (2.12) is the Schwarz Lemma computation from [39,Theorem 4.3]. The second inequality of (2.12) follows from part (i) of Lemma2.2. (cid:3) OLLAPSING LIMITS 15 Two elementary lemmas. In this section, we state and prove twoelementary lemmas which we will need later in our proofs of the collapsingestimates. The first is as follows: Lemma 2.4. Consider a smooth function f : X × [0 , ∞ ) → R which satisfiesthe following conditions: (a) There is a constant A such that for all y ∈ B and all t > |∇ ( f | X y ) | ω | Xy A. (b) For all y ∈ B and all t > we have Z x ∈ X y f ( x, t ) ω n SRF ,y = 0 . (c) There exists a function h : [0 , ∞ ) → [0 , ∞ ) such that h ( t ) → as t → ∞ such that sup x ∈ X f ( x, t ) h ( t ) , for all t > .Then there is a constant C such that sup x ∈ X | f ( x, t ) | C ( h ( t )) / (2 n +1) for all t sufficiently large.Proof. Arguing by contradiction, suppose that there exist t k → ∞ and x k ∈ X with sup X | f ( t k ) | = | f ( x k , t k ) | > k ( h ( t k )) / (2 n +1) , for all k . We may also assume that t k is sufficiently large so that the quantity k ( h ( t k )) / (2 n +1) is less than or equal to 1 for all k . Let y k = π ( x k ), so x k ∈ X y k . Thanks to assumption (c), for all k large, we must have(2.13) f ( x k , t k ) − k ( h ( t k )) / (2 n +1) . This together with assumption (a) implies that for all x in the ω -geodesicball B r ( x k ) in X y k centered at x k and of radius r = k ( h ( t k )) / (2 n +1) A A , we have f ( x, t k ) − k ( h ( t k )) / (2 n +1) . Therefore, using assumptions (b) and (c),0 = Z x ∈ X yk f ( x, t k ) ω n SRF ,y k − k ( h ( t k )) / (2 n +1) Z B r ( x k ) ω n SRF ,y k + Ch ( t k ) . Now the metrics ω SRF ,y k are all uniformly equivalent to each other, and inparticular, Z B r ( x k ) ω n SRF ,y k > C − k n ( h ( t k )) n/ (2 n +1) , using that k ( h ( t k )) / (2 n +1) ω SRF ,y k have volumecomparable to Euclidean balls. Thus we get k n +1 C, which gives a contradiction when k is large. (cid:3) Remark 2.5. Note that Lemma 2.4 also holds with the same proof if X isreplaced by K = π − ( K ′ ) for K ′ a compact subset of B . Hence the resultholds for our more general holomorphic fiber spaces π : X → B as discussedin the introduction, if we restrict to π − ( K ′ ) for a compact set K ′ ⊂ B \ S ′ .The second lemma is an elementary observation from linear algebra. Lemma 2.6. Let A be an N × N positive definite symmetric matrix. Assumethat there exists ε ∈ (0 , with tr A N + ε, det A > − ε. Then there exists a constant C N depending only on N such that k A − I k C N √ ε, where k · k is the Hilbert-Schmidt norm, and I is the N × N identity matrix.Proof. We may assume that N > 2. Let λ , . . . , λ N be the eigenvalues of A .Define the (normalized) elementary symmetric polynomials S k by S k = (cid:18) Nk (cid:19) − X i < ··· − ε . The Maclaurin inequalities S > p S > S /NN , imply that | S − | + | S − | Cε for C depending only on N . Compute k A − I k = N X j =1 ( λ j − = N S − N S − N ( N − S + N C ′ ε, for C ′ depending only on N , and the result follows. (cid:3) OLLAPSING LIMITS 17 A local Calabi estimate for collapsing metrics. Later we willneed a local Calabi estimate for solutions of the K¨ahler-Ricci flow whichare collapsing exponentially in specified directions. The Calabi estimate forthe elliptic complex Monge-Amp`ere equation, giving third order estimatesfor the potential function depending on the second order estimates, firstappeared in the paper of Yau [62], and was used in the parabolic case byCao [6]. In the following we use a formulation due to Phong-Sesum-Sturm[29], together with a cut-off function argument similar to that used in [34]. Proposition 2.7. Let B (0) be the unit polydisc in C n + m and let ω ( n + m ) E = P m + nk =1 √− dz k ∧ dz k be the Euclidean metric. Write C n + m = C m ⊕ C n anddefine ω E,t = ω ( m ) E + e − t ω ( n ) E , for ω ( m ) E and ω ( n ) E Euclidean metrics on thetwo factors C m and C n respectively. Assume that ω = ω ( t ) is a solution ofthe K¨ahler-Ricci flow ∂ω∂t = − Ric( ω ) − ω, ω (0) = ω , on B (0) × [0 , ∞ ) which satisfies (2.14) A − ω E,t ω A ω E,t , for some positive constant A . Then there is a constant C that depends onlyon n, m, A and ω such that for all t > on B / (0) we have (2.15) S = |∇ E g | g Ce t , where g is the metric associated to ω and ∇ E is the covariant derivative of ω ( n + m ) E .Proof. In the following C will denote a generic constant that depends onlyon n, m, A and ω . For simplicity of notation, we write simply ω E = ω ( n + m ) E .Following Yau [62], define S = |∇ E g | g = g iℓ g pk g jq ∇ Ei g jk ∇ Eℓ g pq . If T kij denotes the difference of the Christoffel symbols of ω and ω E , then T isa tensor and it is easy to see that S = | T | g . Since ω E,t is also a flat metric,the quantity S is the same if we use its connection ∇ E,t . The parabolicCalabi computation gives (cfr. [29]) (cid:18) ∂∂t − ∆ (cid:19) S S − |∇ T | − |∇ T | , where here and in the rest of the proof we suppress the subscript g from thenorms.Take ψ a nonnegative smooth cutoff function supported in B (0) with ψ ≡ B / (0) and with √− ∂ψ ∧ ∂ψ Cω E , − Cω E √− ∂∂ ( ψ ) Cω E . From (2.14) it then follows that |∇ ψ | Ce t , ∆( ψ ) > − Ce t . We can then compute (cid:18) ∂∂t − ∆ (cid:19) ( ψ S ) ψ (cid:18) ∂∂t − ∆ (cid:19) S + CSe t + 2 |h∇ ψ , ∇ S i| ψ S − ψ ( |∇ T | + |∇ T | ) + CSe t + 2 |h∇ ψ , ∇ S i| . On the other hand, using the Young inequality2 |h∇ ψ , ∇ S i| = 4 ψ |h∇ ψ, ∇| T | i| ψ |∇ ψ | · |∇| T | | ψ ( |∇ T | + |∇ T | ) + CS |∇ ψ | ψ ( |∇ T | + |∇ T | ) + CSe t , and so (cid:18) ∂∂t − ∆ (cid:19) ( ψ S ) CSe t , (cid:18) ∂∂t − ∆ (cid:19) ( e − t ψ S ) CS. On the other hand, the second order estimate of Cao [6] (the parabolicversion of the estimate of [62, 1]) gives (cid:18) ∂∂t − ∆ (cid:19) tr ω E,t ω = − tr ω E,t ω + e − t g iqE,t g pjE,t ( g ( n ) E ) pq g ij − g iℓE,t g pj g kq ∇ Ei g kj ∇ Eℓ g pq − A − S, using (2.14). It follows that if we take C large enough, then we have (cid:18) ∂∂t − ∆ (cid:19) (cid:0) e − t ψ S + C tr ω E,t ω (cid:1) < . It follows that the maximum of e − t ψ S + C tr ω E,t ω on B (0) × [0 , T ] (forany finite T ) can only occur at t = 0 or on the boundary of B (0), where ψ = 0. Since by (2.14) we have tr ω E,t ω ( n + m ) A , we conclude thatsup B / (0) S Ce t , as required. (cid:3) Collapsing estimates for the K¨ahler-Ricci flow. In this section,we give the proof of Theorem 1.1, which establishes the metric collapsingfor the K¨ahler-Ricci flow in the case of a holomorphic submersion. We willmake use of the lemmas and observations above.We begin with a simple but crucial lemma, which is a generalization of[58, Proposition 7.3]. We will need it later. Lemma 2.8. For any η ∈ (0 , / , there is a constant C such that tr ω ω B − m Ce − ηt . OLLAPSING LIMITS 19 Proof. Let Q = e ηt (tr ω ω B − m ) − e ηt ( ϕ + ˙ ϕ ) . Then from Lemma 2.3, (cid:18) ∂∂t − ∆ (cid:19) Q ηe ηt (tr ω ω B − m ) + Ce ηt − ηe ηt ( ϕ + ˙ ϕ ) − e ηt (tr ω ω B − m ) . From Lemma 2.2, we have tr ω ω B C and | e ηt ( ϕ + ˙ ϕ ) | C. Then (cid:18) ∂∂t − ∆ (cid:19) Q Ce ηt − e ηt (tr ω ω B − m ) , and so at a maximum point of Q we have an upper bound for e ηt (tr ω ω B − m )and hence for Q . The result follows. (cid:3) We will now establish part (ii) of Theorem 1.1. We have the followingestimate in the fibers X y of π . Lemma 2.9. There is a constant C > such that for all t > and all y ∈ B , k e t ω | X y k C ( X y ,ω | Xy ) C, e t ω | X y > C − ω | X y . Proof. Fix a point y ∈ B and a point x ∈ X y . Since π : X → B is aholomorphic submersion, we can find a U ⊂ X a local holomorphic productcoordinate chart for π centered at x , which equals the unit polydisc in C n + m (see e.g. [23, p.60]). Thanks to part (i) of Lemma 2.2, we see that thehypotheses of Proposition 2.7 are satisfied, and so we conclude that on thehalf-sized polydisc we have |∇ E ω | ω Ce t , where ∇ E is the covariant derivative of the Euclidean metric ω E on U .We may assume that ω E is uniformly equivalent to ω on U . Again usingLemma 2.2.(i), we see that on U the metric ω | X y is uniformly equivalent to e − t ω E | X y , and so on U we have (cf. [54, Section 3] or [45, Lemma 3.6.9]) |∇ E ( e t ω | X y ) | ω E = e − t |∇ E ( ω | X y ) | e − t ω E Ce − t |∇ E ( ω | X y ) | ω Ce − t |∇ E ω | ω C. Furthermore, all these estimates are uniform in y ∈ B . The uniform C bound as well as the uniform lower bound for e t ω | X y follows at once. (cid:3) Thanks to Lemma 2.9, for any fixed y ∈ B and 0 < α < 1, given anysequence t k → ∞ we can extract a subsequence such that e t k ω | X y → ω ∞ ,y , in C α on X y , where ω ∞ ,y is a C α K¨ahler metric on X y cohomologous to ω | X y .We can now prove the last statement in Theorem 1.1.(ii). Lemma 2.10. Given any y ∈ B , as t → ∞ we have that e t ω | X y → ω SRF ,y , in C α on X y , for any < α < .Proof. Write( e t ω | X y ) n = e nt ( ω | X y ) n ( ω SRF ,y ) n ( ω SRF ,y ) n = e nt ω n ∧ ω mB ω n SRF ∧ ω mB ( ω SRF ,y ) n = e nt (cid:18) n + mn (cid:19) ω n ∧ ω mB Ω ( ω SRF ,y ) n = e ϕ + ˙ ϕ (cid:18) n + mn (cid:19) ω n ∧ ω mB ω n + m ( ω SRF ,y ) n . Consider the function on X , depending on t , defined by f = e ϕ + ˙ ϕ (cid:18) n + mn (cid:19) ω n ∧ ω mB ω n + m , which when restricted to the fiber X y equals(2.16) f | X y = ( e t ω | X y ) n ( ω SRF ,y ) n . We have that(2.17) Z X y f ( ω SRF ,y ) n = Z X y ( e t ω | X y ) n = Z X y ( ω | X y ) n = Z X y ( ω SRF ,y ) n , so that f − t large we have that(2.18) | e ϕ + ˙ ϕ − | Ce − ηt , as long as η < / 2. On the other hand, if at a point on X y we choosecoordinates so that ω is the identity and ω B is diagonal with eigenvalues λ , . . . , λ m , then at that point(2.19) (cid:18) n + mn (cid:19) ω n ∧ ω mB ω n + m = Y j λ j ( X j λ j /m ) m = (tr ω ω B /m ) m . By Lemma 2.8, sup X tr ω ω B /m Ce − ηt , for η < / 4. Then (2.18) and(2.19) imply that sup X f Ce − ηt , for some η > 0. This shows that f − f − f converges to 1 uniformly on X and exponentially fast as t → ∞ , i.e.(2.20) k ( e t ω | X y ) n − ω n SRF ,y k C ( X y ,ω | Xy ) Ce − ηt , for some η > 0, for all y ∈ B . OLLAPSING LIMITS 21 If we choose a sequence t k → ∞ such that e t k ω | X y → ω ∞ ,y , in C α on X y ,then the C α metric ω ∞ ,y on X y satisfies ω n ∞ ,y = ω n SRF ,y . Standard bootstrap estimates give that ω ∞ ,y is a smooth K¨ahler metric, andthe uniqueness of the solution of this equation (originally due to Calabi)implies that ω ∞ ,y = ω SRF ,y . From this it follows that e t ω | X y → ω SRF ,y in C α , as required. (cid:3) Note that in the previous lemma, since we only have derivative bounds on ω in the fiber directions, we are not able to conclude that the C α convergenceis uniform in y ∈ B . However, we can now establish uniformity of thefiberwise convergence in the C norm and complete the proof of Theorem1.1.(ii). Namely, we will prove the estimate (1.6): k e t ω | X y − ω SRF ,y k C ( X y ,ω | Xy ) Ce − ηt , for all y ∈ B. Proof of (1.6). Consider the function f | X y = ( e t ω | X y ) ∧ ω n − ,y ω n SRF ,y . While this is defined on X y , it is clearly smooth in y and so defines a smoothfunction f on X , which equals f = ( e t ω ) ∧ ω n − ∧ ω mB ω n SRF ∧ ω mB = (cid:18) n + mn (cid:19) ( e t ω ) ∧ ω n − ∧ ω mB Ω= e ϕ + ˙ ϕ (cid:18) n + mn (cid:19) ω ∧ ( e − t ω SRF ) n − ∧ ω mB ω n + m . We wish to show that f converges to 1 uniformly on X and exponentiallyfast. By the arithmetic-geometric means inequality we have( f | X y ) n > ( e t ω | X y ) n ω n SRF ,y , and the right hand side converges uniformly to 1 and exponentially fast by(2.20). Therefore 1 − f satisfies condition (c) in Lemma 2.4. Condition(b) is trivial, and condition (a) holds thanks to Lemma 2.9. Therefore weconclude that(2.21) k ( e t ω | X y ) ∧ ω n − ,y − ω n SRF ,y k C ( X y ,ω | Xy ) Ce − ηt , for some C, η > y ∈ B .Choosing local fiber coordinates at a point x ∈ X y so that ω SRF ,y is theidentity and e t ω | X y is given by the positive definite n × n matrix A we seethat (2.20) and (2.21) give | det A − | Ce − ηt and | tr A − n | Ce − ηt respectively. Applying Lemma 2.6 we obtain k A − I k Ce − ηt/ . Thisimplies the estimate (1.6). (cid:3) We now turn to the proof of part (i) of Theorem 1.1. Proof of Theorem 1.1.(i). Fix a point x ∈ X with y = π ( x ) ∈ B . We have(2.22) tr ω ˜ ω = tr ω ω B + tr ω ( e − t ω SRF ,y ) + e − t ( − tr ω ω B + tr ω ( ω SRF − ω SRF ,y )) . From Lemma 2.2, tr ω ω B is uniformly bounded. We claim that(2.23) | tr ω ( ω SRF − ω SRF ,y ) | Ce t/ . Indeed, since π is a submersion there are coordinates z , . . . , z m + n near x and z , . . . , z m near π ( x ) so that π is given by the map ( z , . . . , z m + n ) ( z , . . . , z m ) (see e.g. [23, p.60]). In particular, z m +1 , . . . , z m + n restrict tolocal coordinates along the fibers of π . Then we can write at x , ω SRF − ω SRF ,y = 2Re √− m X α =1 m + n X j =1 Ψ αj dz α ∧ dz j , for Ψ αj ∈ C . We can do this because the form ω SRF − ω SRF ,y vanishes whenrestricted to the fiber X y . Then | tr ω ( ω SRF − ω SRF ,y ) | = 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re m X α =1 m + n X j =1 g αj Ψ αj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ce t/ , since thanks to estimate in Lemma 2.2.(i) we have that | g jj | Ce t and | g αα | C whenever 1 α m , and so the Cauchy-Schwarz inequality gives | g αj | | g αα g jj | Ce t/ . Moreover, by compactness, we may assume thatthe constant C is independent of the point x and the choice of coordinates.Thus the claim (2.23) is proved.Hence(2.24) | e − t ( − tr ω ω B + tr ω ( ω SRF − ω SRF ,y )) | Ce − t/ . On the other hand, the estimate (1.6) implies that(2.25) tr ω ( e − t ω SRF ,y ) = tr ( e t ω | Xy ) ω SRF ,y n + Ce − ηt . From (2.22), (2.24), (2.25) and Lemma 2.8, we have(2.26) tr ω ˜ ω n + m + Ce − ηt , for some η > 0. Moreover, the constants C and η are independent of thechoice of x ∈ X . On the other hand, for t > T I ,˜ ω n + m ω n + m = ( e − t ω SRF + (1 − e − t ) ω B ) n + m (cid:0) n + mn (cid:1) ω n SRF ∧ ω mB e nt e − ϕ − ˙ ϕ > e − ϕ − ˙ ϕ − Ce − t > − C ′ e − ηt , (2.27)for some η > 0. Here we used Lemma 2.2.(iv) for the last inequality, andthe fact that for k > e − ( n + k ) t ω n + k SRF ∧ ω m − kB ω n SRF ∧ ω mB e nt , OLLAPSING LIMITS 23 is of the order of e − kt .From (2.26) and (2.27) we can apply Lemma 2.6 (choose coordinates sothat ω is the identity and ˜ ω is given by a matrix A ) to obtain k ω − ˜ ω k C ( X,ω ) Ce − ηt , for a uniform C, η > 0. Since ω Cω , we have k ω − ˜ ω k C ( X,ω ) Ce − ηt , and since ˜ ω = ω B + e − t ( ω SRF − ω B ) we obtain the estimate (1.5) as required. (cid:3) Finally, part (iii) of Theorem 1.1 follows from part (i) and the definitionof Gromov-Hausdorff convergence, for example using [58, Lemma 9.1] (notethat our submersion π : X → B is a smooth fiber bundle).3. The K¨ahler-Ricci flow in the case of singular fibers In this section we give the proof of Theorem 1.2.3.1. Preliminaries: the case of singular fibers. Let us first recall thegeneral setup of Song-Tian [39, 40, 42] where our results will apply. Let( X m + n , ω ) be a compact K¨ahler manifold with canonical bundle K X semi-ample and 0 < m := κ ( X ) < dim X .The map π : X → B is a surjective holomorphic map given by sections of H ( X, K ℓX ) where B m is a normal projective variety and the generic fiber X y = π − ( y ) of π has K ℓX y holomorphically trivial (so it is in particular aCalabi-Yau manifold of dimension n ). Recall that we denote by S ′ ⊂ B thesingular set of B together with the set of critical values of π , and we define S = π − ( S ′ ) ⊂ X .Since the map π : X → B ⊂ P H ( X, K ℓX ) is induced by the space ofglobal sections of K ℓX , we have that π ∗ O (1) = K ℓX . Therefore, if we let χ be ℓ ω FS on P H ( X, K ℓX ), we have that π ∗ χ (later, denoted by χ ) is a smoothsemipositive representative of − c ( X ). Here, ω FS denotes the Fubini-Studymetric. We will also denote by χ the restriction of χ to B \ S ′ .As in the submersion case, we can define the semi Ricci-flat form ω SRF on X \ S , so that ω SRF ,y = ω SRF | X y is the unique Ricci-flat K¨ahler metric on X y cohomologous to ω | X y , for all y ∈ B \ S ′ . Let Ω be the volume form on X with √− ∂∂ log Ω = χ, Z X Ω = (cid:18) n + mn (cid:19) Z X ω n ∧ χ m . Define a function F on X \ S by(3.1) F = Ω (cid:0) n + mn (cid:1) ω n SRF ∧ χ m . As in the proof of Theorem 2.1, one sees easily that F is constant along thefibers X y , y ∈ B \ S ′ , so it descends to a smooth function F on B \ S ′ . Then[40, Section 3] shows that the Monge-Amp`ere equation(3.2) ( χ + √− ∂∂v ) m = F e v χ m , has a unique solution v (in the sense of Bedford-Taylor [4]) which is abounded χ -plurisubharmonic function on B , smooth on B \ S ′ . The L ∞ bound for v uses the pluripotential estimates of Ko lodziej [24] (see also thesurvey [30]). We define ω B := χ + √− ∂∂v, which is a smooth K¨ahler metric on B \ S ′ , and satisfies the twisted K¨ahler-Einstein equation Ric( ω B ) = − ω B + ω WP , where ω WP is the smooth semipositive Weil-Petersson form on B \ S ′ con-structed in Section 2.1. We also defineΩ = (cid:18) n + mn (cid:19) ω n SRF ∧ ω mB , and using (3.1) and (3.2) we observe thatΩ = Ω ω mB F χ m = Ω e v , which is thus a bounded strictly positive volume form on X , smooth on X \ S . As in the proof of Theorem 2.1, we can see that √− ∂∂ log Ω = ω B , holds on X \ S .Let then ω = ω ( t ) be a solution of the K¨ahler-Ricci flow ∂∂t ω = − Ric( ω ) − ω, ω (0) = ω , which exists for all time. We gather together some facts about the K¨ahler-Ricci flow in this case which are already known, or which follow easily fromthe existing literature.We begin by writing the flow as a parabolic complex Monge-Amp`ere equa-tion. Since Ω is not smooth everywhere, it is more convenient to use thesmooth volume form Ω . Similarly, since the limiting metric ω B is notsmooth everywhere, we replace it by χ which is smooth on the whole of X .We therefore define the reference metricsˆ ω = e − t ω + (1 − e − t ) χ, which are K¨ahler for all t > 0, and for all t > ω = ˆ ω + √− ∂∂ϕ , and ϕ (0) = 0. Then the K¨ahler-Ricci flow is equivalent to the OLLAPSING LIMITS 25 parabolic complex Monge-Amp`ere equation(3.3) ∂∂t ϕ = log e nt (ˆ ω + √− ∂∂ϕ ) n + m Ω − ϕ, ϕ (0) = 0 , ˆ ω + √− ∂∂ϕ > . It is convenient to introduce, following [54, Section 2], a smooth nonnegativefunction σ on X with zero locus exactly equal to S and with(3.4) σ , √− ∂σ ∧ ∂σ Cχ, − Cχ √− ∂∂σ Cχ, for some constant C (in the case when S is empty, i.e. when π is a submersionand B nonsingular, we can set σ = 1). Explicitly, let I be the ideal sheaf of S ′ inside P H ( X, K ℓX ), let { U j } be an open cover of P H ( X, K ℓX ) such thaton each U j the ideal I is generated by finitely many holomorphic functions { f j,k } , let ρ j be a partition of unity subordinate to this cover, and define asmooth function on P H ( X, K ℓX ) by(3.5) σ = C − X j,k ρ j | f j,k | , where C is a constant chosen so that σ 1. Then the pullback of σ to X is the function that we need. We have the following lemma, which is ananalogue of Lemma 2.2. Lemma 3.1. Let ω = ω ( t ) solve the K¨ahler-Ricci flow as above, and write ϕ = ϕ ( t ) for the solution of (3.3). Then (i) For any compact set K ⊂ X \ S there is a constant C = C ( K ) suchthat C − ˆ ω ω C ˆ ω on K for all t > . (ii) There exists λ > and a positive decreasing function h ( t ) whichtends to zero as t → ∞ such that (3.6) sup X | σ λ ( ϕ − v ) | h ( t ) , for all t > . (iii) There exists C such that | R | C on X × [0 , ∞ ) . (iv) There exists C such that, for h , λ as in (ii), (3.7) sup X | σ λ ( ϕ + ˙ ϕ − v ) | Ch ( t ) , for all t > . Proof. Part (i) is proved in [12] (and is a direct adaptation of [54], see also[39] for the case of elliptic surfaces). Part (ii) is due to Song-Tian (see theproof of [40, Proposition 5.4]). Note that we are free to increase the valueof λ , at the expense of changing the function h ( t ). Part (iii) was proved in[42].The proof of (iv) is almost identical to the proof of Lemma 2.2.(iv). In-deed, it is enough to show that sup X | σ λ ˙ ϕ | Ch ( t ) . We have as before ∂∂t ˙ ϕ = − R − m − ˙ ϕ and hence | ˙ ϕ | , | ∂ ˙ ϕ/∂t | C for some uniform constant C . Suppose for a contradiction that we do not have the desired upper bound of σ λ ˙ ϕ . Then there exists a sequence ( x k , t k ) ∈ X × [0 , ∞ ) with t k → ∞ as k → ∞ such that σ λ ( x k ) ˙ ϕ ( x k , t k ) > kh ( t k ) . In particular, x k S. Put γ k = k C σ λ ( x k ) h ( t k ) . At x k we have σ λ ˙ ϕ > k h ( t k ) on [ t k , t k + γ k ] . Then, using (3.6), we have at x k ,2 h ( t k ) > ( h ( t k + γ k ) + h ( t k )) > σ λ Z t k + γ k t k ˙ ϕdt > γ k k h ( t k ) = k C σ λ h ( t k ) , which gives a contradiction when k → ∞ and we are done. The lower boundis similar. (cid:3) The next lemma follows from a straightforward computation. Lemma 3.2. Along the K¨ahler-Ricci flow, we have on X \ S , (3.8) (cid:18) ∂∂t − ∆ (cid:19) ( ϕ + ˙ ϕ − v ) = tr ω ω B − m. Proof. This follows from the evolution equations (cid:18) ∂∂t − ∆ (cid:19) ϕ = ˙ ϕ − ( n + m ) + tr ω ˆ ω, (cid:18) ∂∂t − ∆ (cid:19) ˙ ϕ = tr ω ( χ − ˆ ω ) + n − ˙ ϕ, and the equation ∆ v = tr ω ( ω B − χ ). (cid:3) Next, from [39, 40, 42], we have: Lemma 3.3. There exist positive constants C, C ′ and λ such that (3.9) χ Cω, ω B Cσ − λ χ, ω B Cσ − λ ω, wherever these quantities are defined, and (3.10) |∇ σ | C, | ∆ σ | C. On X \ S we have (3.11) (cid:18) ∂∂t − ∆ (cid:19) tr ω ω B tr ω ω B + Cσ − λ (tr ω ω B ) C ′ σ − λ . Proof. The first inequality of (3.9) is given in [42, Proposition 2.2]. Thesecond is a consequence of the arguments in [40, Theorem 3.3], as follows:Song-Tian construct the metric ω B by working on a resolution µ : Y → B so that µ − ( S ′ ) is a simple normal crossing divisor E . The class [ µ ∗ χ ] issemipositive and big, and since µ ∗ χ = ℓ ˜ µ ∗ ω FS , where ˜ µ is the composition Y → B → P H ( X, K ℓX ), a standard argument shows that [ µ ∗ χ ] − ε [ E ] OLLAPSING LIMITS 27 contains a K¨ahler metric ω Y , for some ε > 0. Then the results of [9] give abounded solution ˆ v of the Monge-Amp`ere equation(3.12) ( µ ∗ χ + √− ∂∂ ˆ v ) m = ( µ ∗ F ) e ˆ v ( µ ∗ χ ) m , µ ∗ χ + √− ∂∂ ˆ v > , which is smooth away from E . If we let ˆ ω B = µ ∗ χ + √− ∂∂ ˆ v , then ˆ ω B descends to the metric ω B on B \ S ′ . Then [40, Theorem 3.3] givesˆ ω B C | s E | − αh E ω Y , for some α > 0, where s E is a defining section of the line bundle associatedto E and h E is a smooth metric on it. From the construction of σ in (3.5),it follows that µ ∗ σ is bounded above by | s E | βh E for some β > 0. Since µ : Y \ E → B \ S ′ is an isomorphism, the metric ω Y induces a metric ω ′ Y on B \ S ′ and we have(3.13) ω B Cσ − γ ω ′ Y , for some γ > 0. But on Y we also have ω Y C | s E | − δh E µ ∗ χ, for some δ > 0, since the subvariety of Y where ˜ µ fails to be an immersionis contained in E . On B \ S ′ this implies that(3.14) ω ′ Y Cσ − γ ′ χ, for some γ ′ > 0. Combining (3.13) and (3.14) we obtain the second in-equality of (3.9). The third is obtained by combining the first and secondinequalities. The inequalities (3.10) follow from (3.4) and the inequality χ Cω .Local higher order estimates for the equation (3.2) imply that the bisec-tional curvature of ω B is bounded by Cσ − λ on B \ S ′ , up to increasing λ .The Schwarz Lemma calculation [39, Section 4] (see also the exposition in[45, Theorem 3.2.6]) then gives the first inequality (3.11), and the secondfollows from (3.9). (cid:3) Collapsing estimates for the K¨ahler-Ricci flow away from thesingular set. We now give the proof of Theorem 1.2. First, we have ananalogue of Lemma 2.8. It is more complicated in this case because of thesingular set, and the fact that the quantity σ λ ( ϕ + ˙ ϕ − v ), which now takesthe role of ϕ + ˙ ϕ , does not decay exponentially. Lemma 3.4. Given a compact set K ⊂ X \ S , there is a positive decreasingfunction F ( t ) which goes to zero as t → ∞ such that sup K (tr ω ω B − m ) F ( t ) . Proof. Let h ( t ) be a positive decreasing function which goes to zero as t → ∞ such that(3.15) sup X | σ λ ( ϕ + ˙ ϕ − v ) | h ( t ) , which exists thanks to (3.7). We may assume without loss of generality that h ′ ( t ) → t → ∞ . Indeed, by assumption there exists a sequence t i → ∞ with h ( t ) i for all t > t i , and t i +1 − t i > 1. Define a piecewise constantfunction ˜ h ( t ), t > 0, to be equal to i on the interval t i t < t i +1 , so clearly h ( t ) ˜ h ( t ) for all t > t . We can then smooth out ˜ h in the obvious way(making it continuous and essentially linear on each interval t i t < t i +1 ) toobtain a smooth nonnegative and decreasing function ˆ h ( t ) with ˜ h ( t ) ˆ h ( t )for all t > h ( t ) still goes to zero and the derivative of ˆ h ( t ) onthe interval t i t < t i +1 is of the order of ( i ( t i +1 − t i )) − which goes tozero. We then replace h ( t ) by ˆ h ( t ) (still calling it h ( t )).Next, we pick a smooth positive function ℓ ( t ) , t > 0, with lim t →∞ ℓ ( t ) = ∞ , such that | ℓ ′ ( t ) | C for all t > ℓ ( t ) h ( t ) , ℓ ( t ) − h ′ ( t ) . Define Q = ℓ ( t ) σ λ (tr ω ω B − m ) − h ( t ) σ λ ( ϕ + ˙ ϕ − v ) . Then using Lemmas 3.2, 3.3 and (3.15), we have on X \ S , (cid:18) ∂∂t − ∆ (cid:19) Q (cid:18) ℓ ′ ( t ) − h ( t ) (cid:19) σ λ (tr ω ω B − m ) + C ( ℓ ( t ) + 1) + h ′ ( t ) h ( t ) σ λ ( ϕ + ˙ ϕ − v ) − (cid:28) ∇ ( σ λ ) , ∇ (cid:18) ℓ ( t )(tr ω ω B − m ) − h ( t ) ( ϕ + ˙ ϕ − v ) (cid:19)(cid:29) (cid:18) ℓ ′ ( t ) − h ( t ) (cid:19) σ λ (tr ω ω B − m ) + C ( ℓ ( t ) + 1) − h ′ ( t ) h ( t ) − σ − λ Re D ∇ ( σ λ ) , ∇ Q E + CQσ − , where we have used the fact that | ∆( σ λ ) | Cσ λ . Observe that from (3.9)and (3.15) we have σ − Q Cℓ ( t ), since we may assume that λ > 1. Since | ℓ ′ ( t ) | C , we may assume that t is sufficiently large so that ℓ ′ ( t ) h ( t ) . We wish to obtain an upper bound for Q using the maximum principle.Hence we may assume without loss of generality that we are working at apoint with tr ω ω B − m > 0. Therefore we get (cid:18) ∂∂t − ∆ (cid:19) Q − h ( t ) σ λ (tr ω ω B − m ) + C ( ℓ ( t ) + 1) − h ′ ( t ) h ( t ) − σ − λ Re D ∇ ( σ λ ) , ∇ Q E , OLLAPSING LIMITS 29 and so at a maximum point of Q (which is necessarily in X \ S ) we have Q Cℓ ( t )( ℓ ( t ) + 1) h ( t ) − ℓ ( t ) h ′ ( t ) + C C, thanks to our choice of ℓ ( t ). This proves what we want, choosing F ( t ) = Cℓ ( t ) , for C depending on the compact set K . (cid:3) The rest of the proof of Theorem 1.2 is almost identical to the proof ofTheorem 1.1. Proof of Theorem 1.2. To avoid repetition, we provide here just an outlineof the proof, emphasizing the changes from the proof of Theorem 1.1.For the rest of the proof, fix a compact set K ′ ⊂ B \ S ′ and write K = π − ( K ′ ) ⊂ X \ S .First, there exists a constant C such that k e t ω | X y k C ( X y ,ω | Xy ) C, e t ω | X y > C − ω | X y , for all t > y ∈ K ′ . Indeed, this follows from the same argumentas in Lemma 2.9, since π is a submersion near every point in π − ( K ′ ).In particular, for any fixed y ∈ B \ S ′ and 0 < α < 1, given any sequence t k → ∞ we can extract a subsequence such that(3.16) e t k ω | X y → ω ∞ ,y , in C α on X y , where ω ∞ ,y is a C α K¨ahler metric on X y cohomologous to ω | X y .Next we claim that we have fiberwise C α convergence of e t ω | X y to ω SRF ,y for y ∈ K ′ . More precisely, given α ∈ (0 , 1) and y ∈ K ′ , we have that(3.17) e t ω | X y → ω SRF ,y , in C α as t → ∞ on X y .Indeed, on X \ S write f = e ϕ + ˙ ϕ − v (cid:18) n + mn (cid:19) ω n ∧ ω mB ω n + m , so that restricted to X y ( y ∈ K ′ ) we have ( e t ω | X y ) n = f ( ω SRF ,y ) n . Notethat Z X y ( f − ω SRF ,y ) n = 0 , and, thanks to (3.7),sup K | e ϕ + ˙ ϕ − v − | → , as t → ∞ . Moreover, as in (2.19) and using Lemma 3.4, on K we have (cid:18) n + mn (cid:19) ω n ∧ ω mB ω n + m (tr ω ω B /m ) m H ( t ) , for H ( t ) → t → ∞ . Hence f − h ( t ) , with h ( t ) → t → ∞ .Finally, f satisfies |∇ ( f | X y ) | ω | Xy A for all y ∈ K ′ . Applying Lemma 2.4 to f − K (see Remark 2.5) we have that f converges to 1 uniformly on K . Namely,(3.18) k ( e t ω | X y ) n − ω n SRF ,y k C ( X y ,ω | Xy ) → , as t → ∞ , uniformly as y varies in K ′ . In particular, e t ω | X y → ω SRF ,y in C α by (3.16)and the same argument as in Lemma 2.10.Next we show that(3.19) k e t ω | X y − ω SRF ,y k C ( X y ,ω | Xy ) → , as t → ∞ , uniformly for y ∈ K ′ . To see this, define f = e ϕ + ˙ ϕ − v (cid:18) n + mn (cid:19) ω ∧ ( e − t ω SRF ) n − ∧ ω mB ω n + m , which satisfies f | X y = ( e t ω | X y ) ∧ ω n − ,y ω n SRF ,y , when restricted to X y , for y ∈ K ′ . By the same argument as in the proof of(1.6) in Section 2, we see that f converges 1 as t → k ( e t ω | X y ) ∧ ω n − ,y − ω n SRF ,y k C ( X y ,ω | Xy ) → , as t → ∞ , uniformly as y varies in K ′ . From (3.18) and (3.20), we apply Lemma 2.6to obtain (3.19) as required.It remains to prove part (i) of Theorem 1.2. Define˜ ω = e − t ω SRF + (1 − e − t ) ω B , as in Section 2. On K we havetr ω ˜ ω = tr ω ω B + tr ω ( e − t ω SRF ,y ) + e − t ( − tr ω ω B + tr ω ( ω SRF − ω SRF ,y )) . On K , tr ω ω B is uniformly bounded. It then follows by the same argumentas in the proof of (2.23) that(3.21) sup K | e − t ( − tr ω ω B + tr ω ( ω SRF − ω SRF ,y )) | Ce − t/ . Moreover, (3.19) implies that tr ω ( e − t ω SRF ,y ) n + h ( t ) for h ( t ) → 0, on K .Then from Lemma 3.4 we have on K , after possibly changing h ( t ),(3.22) tr ω ˜ ω − ( n + m ) h ( t ) → . On the other hand, again on K ,˜ ω n + m ω n + m = (cid:18) n + mn (cid:19) e − nt ω n SRF ∧ ω mB ω n + m + O ( e − t ) = e − ϕ − ˙ ϕ + v + O ( e − t ) → . Applying Lemma 2.6, we obtain k ω − ˜ ω k C ( K,ω ) → , as t → ∞ . But since ω Cω on K , and ˜ ω = ω B + e − t ( ω SRF − ω B ) we obtain k ω − ω B k C ( K,ω ) → , as t → ∞ , OLLAPSING LIMITS 31 as required. (cid:3) Collapsing of Ricci-flat K¨ahler metrics In this section we prove Theorem 1.3.4.1. Monge-Amp`ere equations and preliminary estimates. Let ( X, ω X )be a compact ( n + m ) manifold with a Ricci-flat K¨ahler metric ω X , as inSection 1.3 of the Introduction. Recall that we have a holomorphic map π : ( X, ω X ) → ( Z, ω Z ) between compact K¨ahler manifolds with (possiblysingular) image B , a normal variety in Z . We denote by S ′ ⊂ B thosepoints of B which are either singular in B or critical for π , and we write S = π − ( S ′ ) . The fibers X y for y ∈ B \ S ′ are Calabi-Yau n -folds.Write χ = π ∗ ω Z , which is a smooth nonnegative (1 , 1) form on X , andwe will also write χ for the restriction of ω Z to B \ S ′ . Note that R B \ S ′ χ m isfinite.We define a semi Ricci-flat form ω SRF on X \ S in the same way as inSection 2.1. Indeed, for each y ∈ B \ S ′ there is a smooth function ρ y on X y so that ω X | X y + √− ∂∂ρ y = ω SRF,y is Ricci-flat, normalized by R X y ρ y ( ω X | X y ) n = 0. As y varies, this defines a smooth function ρ on X \ S and we define ω SRF = ω X + √− ∂∂ρ. Let F be the function on X \ S given by F = ω n + mX (cid:0) n + mn (cid:1) ω n SRF ∧ χ m . As in the proof of Theorem 2.1, one sees easily that F is constant along thefibers X y , y ∈ B \ S ′ , so it descends to a smooth function F on B \ S ′ . Wesee that F satisfies R B \ S ′ F χ m = R X ω n + mX / (cid:0) n + mn (cid:1) R X y ω nX (see [40, Section3] and [54, Section 4]). Here note that R X y ω nX is independent of y ∈ B \ S ′ .Then [40, Section 3] shows that the Monge-Amp`ere equation(4.1) ( χ + √− ∂∂v ) m = (cid:0) n + mn (cid:1) R X ω nX ∧ χ m R X ω n + mX F χ m , has a unique solution v (in the sense of Bedford-Taylor [4]) which is abounded χ -plurisubharmonic function on B , smooth on B \ S ′ , with R X vω n + mX =0, where here and henceforth we write v for π ∗ v .We define ω B = χ + √− ∂∂v, for v solving (4.1). Note that we have(4.2) ω n SRF ∧ ω mB = (cid:0) n + mn (cid:1) R X ω nX ∧ χ m R X ω n + mX F ω n SRF ∧ χ m = R X ω nX ∧ χ m R X ω n + mX ω n + mX . Moreover, ω B is a smooth K¨ahler metric on B \ S ′ , and satisfiesRic( ω B ) = ω WP , where ω WP is a smooth semipositive form on B \ S ′ constructed in the sameway as in Section 2.1.As in Section 3.1, we fix a smooth nonnegative function σ on X with zerolocus exactly equal to S and with(4.3) σ , √− ∂σ ∧ ∂σ Cχ, − Cχ √− ∂∂σ Cχ, for some constant C (in the case when S is empty we set σ = 1). It isconvenient to define another smooth nonnegative function F with zero locusequal to S by(4.4) F = e − e Aσ − λ , for positive constants A and λ to be determined.For t > 0, let ˜ ω = ˜ ω ( t ), given by˜ ω = χ + e − t ω X ∈ α t = [ χ ] + e − t [ ω X ] , be a family of reference K¨ahler metrics, and let ω = ˜ ω + √− ∂∂ϕ be theunique Ricci-flat K¨ahler metric on X cohomologous to ˜ ω , with the normal-ization R X ϕω n + mX = 0. Then ω solves the Calabi-Yau equation(4.5) ω n + m = c t e − nt ω n + mX , where c t is the constant given by(4.6) c t = R X e nt ˜ ω n + m R X ω n + mX = 1 R X ω n + mX m X k =0 (cid:18) n + mk (cid:19) e − ( m − k ) t Z X ω n + m − kX ∧ χ k , so(4.7) c t = (cid:18) n + mn (cid:19) R X ω nX ∧ χ m R X ω n + mX + O ( e − t ) . The following estimates are already known by the work of the first-namedauthor [54]. Lemma 4.1. Let ω = ω ( t ) and ϕ = ϕ ( t ) be as above. Then (i) There exist positive constants C, A, λ such that for F given by (4.4), (4.8) C − F ˜ ω ω C F − ˜ ω, for all t > . (ii) There exists C such that sup X | ϕ | C for all t > . (iii) Z X | ϕ − v | ω n + mX → as t → ∞ . (iv) For any α ∈ (0 , , we have ϕ → v in C ,α loc ( X \ S ) as t → ∞ .Proof. Part (i) is proved in [54] (see also [18, Lemma 4.1]). For part (ii), see[54, Theorem 2.1]. Parts (iii) and (iv) are proved in [54, Theorem 4.1]. (cid:3) The estimate (4.8) is the reason why here and in the rest of this sectionwe consider the function F instead of the simpler σ − λ which we used in thecase of the K¨ahler-Ricci flow. We do not expect this to be optimal.The next lemma is analogous to Lemma 3.3 above. OLLAPSING LIMITS 33 Lemma 4.2. There exist positive constants C, C ′ and λ such that (4.9) χ Cω, ω B Cσ − λ χ, ω B Cσ − λ ω, wherever these quantities are defined, and (4.10) |∇ σ | C, | ∆ σ | C. On X \ S we have (4.11) ∆tr ω ω B > − tr ω ω B − Cσ − λ (tr ω ω B ) > − C ′ σ − λ . Proof. The proof is almost the same as that of Lemma 3.3. The first inequal-ity of (4.9) is given by the Schwarz Lemma [54, Lemma 3.1]. The secondfollows again from the arguments in [40, Theorem 3.3], as in Lemma 3.3 (thefact that now B is not necessarily projective does not affect the arguments),and the third follows immediately. The inequalities (4.10) then follow from(4.3).Local higher order estimates for the equation (4.1) imply that the bisec-tional curvature of ω B is bounded by Cσ − λ on B \ S ′ , up to increasing λ .The Schwarz Lemma calculation [54, Lemma 3.1] gives the first inequality(4.11), and the second follows from (4.9). (cid:3) Collapsing estimates for Ricci-flat metrics. In this section, wegive the proof of Theorem 1.3. Recall that the function F is given by (4.4)and depends on the two constants A and λ . Lemma 4.3. There are constants A, λ > and a positive function h ( t ) → as t → ∞ , such that (4.12) sup X F | ϕ − v | h ( t ) . Proof. We choose A, λ as in Lemma 4.1.(1). Taking the trace with respectto g X of ω = ˜ ω + √− ∂∂ϕ and applying Lemma 4.1.(i), we have(4.13) sup X F | ∆ g X ϕ | C. From Lemma 4.1.(ii) and the fact that v is bounded we have(4.14) sup X | ϕ | + sup X | v | C. It follows that(4.15) sup X F | ∂ϕ | g X C, from a rather standard interpolation type argument. Indeed, note thatthe Sobolev imbedding theorem and the L p elliptic estimates give for p > n + m ),sup X | ∂ ( F ϕ ) | g X C (cid:18)Z X | ∆ g X ( F ϕ ) | p (cid:19) /p + (cid:18)Z X |F ϕ | p (cid:19) /p ! , where we are integrating with respect to the volume form of g X . Using(4.13), (4.14) and the Cauchy-Schwarz inequality, we obtainsup X F | ∂ϕ | g X C (cid:18)Z X ( | ∂ϕ | g X | ∂ F | g X ) p (cid:19) /p + 1 ! C sup X ( F | ∂ϕ | g X ) ( p − /p (cid:18)Z X | ∂ϕ | g X ˆ F (cid:19) /p + 1 ! , (4.16)where ˆ F = | ∂ F | pg X / F p − . From the definition of F we see that ˆ F + | ∂ ˆ F | g X C F and hence, integrating by parts, Z X | ∂ϕ | g X ˆ F = − Z X (cid:16) ϕ (∆ g X ϕ ) ˆ F − ϕ h ∂ϕ, ∂ ˆ F i g X (cid:17) C (sup X F | ∂ϕ | g X + 1) . (4.17)Then (4.15) follows from (4.16) and (4.17).Next, the estimates in [40, Section 3], together with a similar argumentgive(4.18) sup X F | ∂v | g X C. Combining (4.14), (4.15) and (4.18), we obtain(4.19) sup X | ∂ ( F ( ϕ − v )) | g X C. From Lemma 4.1.(iii),(4.20) Z X F | ϕ − v | ω n + mX C Z X | ϕ − v | ω n + mX → , as t → ∞ , and (4.12) now follows from (4.19) and (4.20). (cid:3) Write ˙ ϕ and ¨ ϕ for the first and second t -derivatives of ϕ . Then: Lemma 4.4. There is a uniform constant C so that (4.21) sup X | ˙ ϕ | C, and (4.22) sup X ¨ ϕ C. Proof. First, we have ∆ ϕ = n + m − tr ω ˜ ω. Differentiating the logarithm of (4.5) with respect to t , we obtain(4.23) (log c t ) ′ − n = tr ω ( √− ∂∂ ˙ ϕ + χ − ˜ ω ) , and so ∆ ˙ ϕ = − n + (log c t ) ′ + tr ω ˜ ω − tr ω χ. OLLAPSING LIMITS 35 Hence(4.24) ∆( ϕ + ˙ ϕ ) = m − tr ω χ + (log c t ) ′ . We have that (log c t ) ′ = O ( e − t ), because thanks to (4.6) we can writelog c t = log m X k =1 a k e − kt ! + const , for some positive constants a k , and the claim follows from differentiating thisexpression. Taking one more derivative, we also see that (log c t ) ′′ = O ( e − t ) . From the Schwarz Lemma (4.9) we have sup X tr ω χ C , and hence | ∆( ϕ +˙ ϕ ) | C . The normalization for ϕ implies that R X ˙ ϕω n + mX = 0. We nowcompute the Laplacian of ¨ ϕ , by taking a derivative of (4.23) to get∆ ¨ ϕ = (log c t ) ′′ + tr ω χ − tr ω ˜ ω + |√− ∂∂ ˙ ϕ + χ − ˜ ω | g , and so ∆( ˙ ϕ + ¨ ϕ ) > − C and R X ¨ ϕω n + mX = 0.The Ricci-flat metrics ω have a uniform upper bound on their diameter,thanks to [53] (and independently [64]), and have volume bounded below by C − e − nt , by (4.5). Therefore the Green’s function G ( x, y ) of the Laplacianof ∆ (normalized by R y ∈ X G ( x, y ) ω n + m ( y ) = 0), satisfies G ( x, y ) > − Ce nt , for a uniform constant C (see e.g. [2, Theorem 3.2] or [35, Chapter 3,Appendix A]). Also, again from (4.5), Z X ( ϕ + ˙ ϕ ) ω n + m = c t e − nt Z X ( ϕ + ˙ ϕ ) ω n + mX = 0 , and so the Green’s formula for the metric ω and the bound | ∆( ϕ + ˙ ϕ ) | C imply thatsup X ( ϕ + ˙ ϕ ) = − Z y ∈ X ∆( ϕ + ˙ ϕ )( y ) G ( x, y ) ω n + m ( y )= Z y ∈ X ( − ∆( ϕ + ˙ ϕ )( y ))( G ( x, y ) + Ce nt ) ω n + m ( y ) Ce nt Z X ω n + m C, where x is any point where ϕ + ˙ ϕ achieves its maximum. Similarly we get alower bound for ϕ + ˙ ϕ , and so we have shown(4.25) sup X | ϕ + ˙ ϕ | C, which, together with sup X | ϕ | C (Lemma 4.1.(ii)), proves (4.21).Next, since ∆( ˙ ϕ + ¨ ϕ ) > − C and R X ( ˙ ϕ + ¨ ϕ ) ω n + mX = 0, exactly the sameargument shows that sup X ( ˙ ϕ + ¨ ϕ ) C, which together with (4.21) proves (4.22). (cid:3) Remark 4.5. One can feed the bound (4.25) into a Cheng-Yau type argu-ment, exactly as in [37, Lemma 3.6], by applying the maximum principleto Q = | ∂ ( ϕ + ˙ ϕ ) | g A − ϕ − ˙ ϕ + tr ω χ, where A is chosen so that A − ϕ − ˙ ϕ > A/ > 0, and prove thatsup X | ∂ ( ϕ + ˙ ϕ ) | g C. However, we won’t need this estimate.The quantity ( ϕ + ˙ ϕ − v ) will play here the same role as in Section 3.Indeed, observe that from (4.24),(4.26) ∆( ϕ + ˙ ϕ − v ) = m − tr ω ω B + (log c t ) ′ , and (log c t ) ′ = O ( e − t ) (cf. (3.8)). The following lemma is analogous to part(iv) of Lemma 3.1. Lemma 4.6. There is a positive function H ( t ) with H ( t ) → as t → ∞ such that (4.27) sup X F | ϕ + ˙ ϕ − v | H ( t ) . Proof. First we prove that lim sup t > sup X F ( ϕ + ˙ ϕ − v ) 0. Thanks to(4.12) it is enough to show thatlim sup t > sup X F ˙ ϕ . If this is not the case, then there exist ε > x k ∈ X and t k → ∞ such thatsup X F ˙ ϕ ( t k ) = F ( x k ) ˙ ϕ ( t k , x k ) > ε, so in particular x k S . From Lemma 4.4 we see that(4.28) ∂∂t ( F ˙ ϕ ) = F ¨ ϕ C, and so F ( x k ) ˙ ϕ ( t, x k ) > ε , for t ∈ [ t k − ε C , t k ] . Integrating over t , F ( x k )( ϕ − v )( t k , x k ) > F ( x k )( ϕ − v )( t k − ε/ C, x k ) + ε C , but from (4.12) we have h ( t k ) > − h ( t k − ε/ C ) + ε C , for a positive function h ( t ) with h ( t ) → t → ∞ . Letting k → ∞ we geta contradiction.The inequality lim inf t > inf X F ( ϕ + ˙ ϕ − v ) > ϕ , we can insteadreplace the time interval [ t k − ε C , t k ] by [ t k , t k + ε C ]. (cid:3) OLLAPSING LIMITS 37 We can finally prove the following analogue of Lemma 3.4: Lemma 4.7. Given a compact set K ⊂ X \ S , there is a positive decreasingfunction F ( t ) which goes to zero as t → ∞ such that sup K (tr ω ω B − m ) F ( t ) . Proof. For the purpose of this proof, define a function ˜ F = e − e ˜ Aσ − λ , with ˜ A slightly larger than the constant A of F used in (4.27). We will apply themaximum principle to Q = ˜ F (tr ω ω B − m ) − ˜ F p H ( t ) ( ϕ + ˙ ϕ − v ) , where H ( t ) is the function from (4.27).We may assume, by increasing ˜ A if necessary, that(4.29) |∇ ˜ F | ˜ F C F , | ∆ ˜ F | C F , where we used (4.3) and the Schwarz Lemma estimate tr ω χ C . It followsfrom Lemma 4.6 that(4.30) | ( ϕ + ˙ ϕ − v )∆ ˜ F | CH ( t )and(4.31) | ˜ F ( ϕ + ˙ ϕ − v ) | |∇ ˜ F | ˜ F CH ( t ) . Then we compute, using (4.26) and (4.30),∆( ˜ F ( ϕ + ˙ ϕ − v )) − ˜ F (tr ω ω B − m − (log c t ) ′ ) + 2Re h∇ ˜ F , ∇ ( ϕ + ˙ ϕ − v ) i + CH ( t ) . From Lemma 4.2, we have, for some λ ′ > F (tr ω ω B − m )) > − C ˜ F σ − λ ′ + (tr ω ω B − m )∆ ˜ F + 2Re h∇ ˜ F , ∇ (tr ω ω B − m ) i > h∇ ˜ F , ∇ (tr ω ω B − m ) i − C. Hence∆ Q > ˜ F p H ( t ) (tr ω ω B − m − (log c t ) ′ ) − C + 2Re h∇ ˜ F , ∇ ( Q/ ˜ F ) i = ˜ F p H ( t ) (tr ω ω B − m − (log c t ) ′ ) − C + 2˜ F Re h∇ ˜ F , ∇ Q i − Q ˜ F |∇ ˜ F | > ˜ F p H ( t ) (tr ω ω B − m − (log c t ) ′ ) − C + 2˜ F Re h∇ ˜ F , ∇ Q i , where in the last line we used (4.31). At the maximum of Q we have ∇ Q = 0.We may assume without loss of generality that (log c t ) ′ / p H ( t ) → t → ∞ . Making use of Lemma 4.6, we obtain Q C p H ( t ). The resultfollows. (cid:3) Another ingredient that we will need is the following local Calabi estimate,whose proof can be obtained by a straightforward modification of the proofof Proposition 2.7. Proposition 4.8. Let B (0) be the unit polydisc in C n + m and let ω ( n + m ) E = P m + nk =1 √− dz k ∧ dz k be the Euclidean metric. Write C n + m = C m ⊕ C n anddefine ω E,t = ω ( m ) E + e − t ω ( n ) E . Assume that ω = ω ( t ) is a Ricci-flat K¨ahlermetric which satisfies for t > , (4.32) A − ω E,t ω A ω E,t , for some positive constant A . Then there is a constant C that depends onlyon n, m, A such that for all t > on B / (0) we have (4.33) S = |∇ E g | g Ce t , where g is the metric associated to ω and ∇ E is the covariant derivative of ω ( n + m ) E . We can now complete the proof of Theorem 1.3. Proof of Theorem 1.3. The proof is similar to the proof of Theorem 1.2, sowe provide here just a sketch. Fix a compact set K ′ ⊂ B \ S and write K = π − ( K ′ ). From Lemma 4.1.(i) and Proposition 4.8, as in the case ofthe flow, there is a constant C such that(4.34) k e t ω | X y k C ( X y ,ω X | Xy ) C, e t ω | X y > C − ω X | X y , for all t > y ∈ K ′ .On X \ S , write for any y ∈ K ′ ,( e t ω | X y ) n = e nt ω n ∧ ω mB ω n SRF ∧ ω mB ( ω SRF ,y ) n = f ( ω SRF ,y ) n , for f = c t R X ω n + mX R X ω nX ∧ χ m ω n ∧ ω mB ω n + m , where we used (4.2) and (4.5). Note that when restricted to the fiber, f isgiven by(4.35) f | X y = ( e t ω | X y ) n ( ω SRF ,y ) n . We have that(4.36) Z X y f ( ω SRF ,y ) n = Z X y ( e t ω | X y ) n = Z X y ( ω X | X y ) n = Z X y ( ω SRF ,y ) n , so that f − c t R X ω n + mX R X ω nX ∧ χ m → (cid:18) n + mn (cid:19) , OLLAPSING LIMITS 39 as t → ∞ . Moreover, as in (2.19), using Lemma 4.7, (cid:18) n + mn (cid:19) ω n ∧ ω mB ω n + m (tr ω ω B /m ) m H ( t ) , for H ( t ) → t → ∞ . Hence f − h ( t ) , with h ( t ) → t → ∞ .Finally, f satisfies |∇ ( f | X y ) | ω X | Xy A for all y ∈ K ′ . Applying Lemma 2.4to f − K (see Remark 2.5), f converges to 1 uniformly on K . Namely,(4.38) k ( e t ω | X y ) n − ω n SRF ,y k C ( X y ,ω X | Xy ) → , as t → ∞ , uniformly as y varies in K ′ . It follows that for any given y ∈ K ′ we have e t ω | X y → ω SRF ,y , as t → ∞ , in C α on X y , for any α ∈ (0 , k e t ω | X y − ω SRF ,y k C ( X y ,ω X | Xy ) → , as t → ∞ , uniformly for y ∈ K ′ . To see this, define f | X y = ( e t ω | X y ) ∧ ω n − ,y ω n SRF ,y . While this is defined on X y , it is clearly smooth in y and so defines a smoothfunction f on X \ S , which equals f = e t ω ∧ ω n − ∧ ω mB ω n SRF ∧ ω mB = c t R X ω n + mX R X ω nX ∧ χ m ω ∧ ( e − t ω SRF ) n − ∧ ω mB ω n + m . By the same argument as in the proof of (1.6) in Section 2, we see that f converges 1 as t → k ( e t ω | X y ) ∧ ω n − ,y − ω n SRF ,y k C ( X y ,ω X | Xy ) → , as t → ∞ , uniformly as y varies in K ′ . From (4.38) and (4.40), we apply Lemma 2.6to obtain (4.39) as required.It remains to prove part (i) of Theorem 1.3. Define ˆ ω = ˆ ω ( t ) byˆ ω = e − t ω SRF + (1 − e − t ) ω B . By the same argument as in the proof of Theorem 1.2.(i), we have on K ,(4.41) tr ω ˆ ω − ( n + m ) h ( t ) → , for a positive decreasing function h ( t ) (depending on K ). 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