Collapsing of abelian fibered Calabi–Yau manifolds

COLLAPSING OF ABELIAN FIBEREDCALABIYAU MANIFOLDS

MARK GROSS, VALENTINO TOSATTI, and YUGUANG ZHANG

AbstractWe study the collapsing behavior of Ricci-flat Khler metrics on a projective CalabiYau manifold which admits an abelian fibration, when the volume of the fibersapproaches zero. We show that away from the critical locus of the fibration the met-rics collapse with locally bounded curvature, and along the fibers the rescaled metricsbecome flat in the limit. The limit metric on the base minus the critical locus is locallyisometric to an open dense subset of any GromovHausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families ofpolarized hyperkhler manifolds in the large complex structure limit. In this setting,we prove an analogue of a result of Gross and Wilson for K3 surfaces, which is moti-vated by the StromingerYauZaslow picture of mirror symmetry.

1. IntroductionIn this article, a CalabiYau manifold M is a compact Khler manifold with vanish-ing first Chern class c1.M/ D 0 in H 2.M;R/. A fundamental theorem of Yau [43,Theorem 2] says that on M there exists a unique Ricci-flat Khler metric in eachKhler class. If we move the Khler class towards a limit class on the boundary ofthe Khler cone, we get a family of Ricci-flat Khler metrics which degenerates inthe limit. The general question of understanding the geometric behavior of these met-rics was raised by Yau [44], [45], Wilson [42], and others, and much work has beendevoted to it (see, e.g., [17], [28], [29], [33], [35], [36] and references therein). Inthis article, we study metric degenerations of Ricci-flat Khler metrics whose Khlerclasses approach semiample, nonbig classes.

The first useful observation to make is that the diameters of a family of Ricci-flatKhler metrics Q!t , t 2 .0; 1, on a CalabiYau manifold M , are uniformly boundedif their Khler classes Q!t tend to a limit class on the boundary of the Khler coneDUKE MATHEMATICAL JOURNALVol. 162, No. 3, 2013 DOI 10.1215/00127094-2019703Received 9 August 2011. Revision received 17 April 2012.2010 Mathematics Subject Classification. Primary 32Q25; Secondary 32W20, 14J32, 14J33, 53C26.Grosss work partially supported by National Science Foundation grants DMS-0805328 and DMS-1105871.Tosattis work partially supported by National Science Foundation grant DMS-1005457.Zhangs work partially supported by National Science Foundation of China grant NSFC-10901111.

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518 GROSS, TOSATTI, and ZHANG

when t ! 0 (see [35], [46]). Another special feature of the Khler case is that the vol-ume of the Ricci-flat metrics can be computed cohomologically; to determine whetherit will approach zero or stay bounded away from zero, it is enough to calculate theself-intersection n where n D dimCM . If n is strictly positive, then it was provedby the second author in [35] that the Ricci-flat metrics do not collapse (i.e., there is aconstant > 0 independent of t such that each Q!t has a unit radius metric ball withvolume bigger than ), and in fact they converge smoothly away from a subvariety.If n is zero, then the total volume of the Ricci-flat metrics approaches zero, so oneexpects to have collapsing to a lower-dimensional space. This was shown to be thecase for elliptically fibered K3 surfaces by Gross and Wilson [17], and later the sec-ond author considered the higher-dimensional case when the CalabiYau manifoldM admits a holomorphic fibration to a lower-dimensional Khler space, and the limitclass is the pullback of a Khler class (see [36]). The first goal of this paper is toimprove the convergence result in [36].

Let us now describe our first result in detail. Let .M;!M / be a compact CalabiYau n-manifold which admits a holomorphic map f W M ! Z, where .Z;!Z/ isa compact Khler manifold. Thanks to Yaus theorem, we can assume that !M isRicci-flat. Denote by N D f .M/ the image of f , and assume that N is an irreduciblenormal subvariety of Z with dimension m, 0 < m < n, and assume that the map f WM ! N has connected fibers. Define !0 WD f !Z , which is a smooth nonnegativereal .1; 1/-form on M with cohomology class on the boundary of the Khler cone,and let !N be the restriction of !Z to the regular part of N . For example, one cantake either Z D N (if N is smooth) or Z D CPk (if N is an algebraic variety). Thissecond case arises if we have a line bundle L ! M which is semiample (some poweris globally generated) and of Iitaka dimension m< n, so L is not big.

In general, given a map f W M ! N as above, there is a proper analytic sub-variety S M such that Nnf .S/ is smooth and such that f W MnS ! Nnf .S/ isa smooth submersion (the set f .S/ is exactly the image of the subset of M wherethe differential df does not have full rank m). For any y 2 Nnf .S/, the fiber My Df 1.y/ is a smooth CalabiYau manifold of dimension nm, and it is equipped withthe Khler metric !M jMy . The volume of the fibers

RMy

.!M jMy /nm is a homo-logical constant which we can assume to be 1. Consider the Khler metrics on Mgiven by !t D !0 C t!M , with 0 < t 1, and call Q!t D !t C

p1@N@'t the uniqueRicci-flat Khler metric on M cohomologous to !t , with potentials normalized bysupM 't D 0. They satisfy a family of complex MongeAmpre equations

Q!nt D .!t Cp1@N@'t /n D ct tnm!nM ; (1.1)

where ct is a constant that has a positive limit as t ! 0 (see (4.15)). A generalC 0-estimate k'tkC0 C (independent of t > 0) for such equations was proved byDemailly and Pali [9] and by Eyssidieux, Guedj, and Zeriahi [10], generalizing work

COLLAPSING OF ABELIAN FIBERED CALABIYAU MANIFOLDS 519

of Koodziej [23]. In the case under consideration, much more is true: the secondauthors work [36] shows that there exists a smooth function ' on Nnf .S/ so thatas t goes to zero we have 't ! ' f in C 1;loc .MnS;!M / for any 0 < < 1. More-over, ! D !N C

p1@N@' is a Khler metric on Nnf .S/ with Ric.!/ D !WP. Here!WP is the pullback of the WeilPetersson metric from the moduli space of polar-ized CalabiYau fibers, which has appeared several times before in the literature (see[11], [17], [33], [36]).

We now assume that one (and hence every) fiber My with y 2 Nnf .S/ is biholo-morphic to a complex torus. This is the case for example whenever M is hyperkhler.We also assume that M is projective, so we can take !M to be the first Chern classof an ample line bundle. In this case, we can improve the above result, thus answeringQuestions 4.1 and 4.2 of [37] in our setting, as follows.

THEOREM 1.1If M is projective and if one (and hence all) of the fibers My with y 2 Nnf .S/ is atorus, then as t approaches zero the Ricci-flat metrics Q!t converge in C1loc .MnS;!M /to f !, where ! is a Khler metric on Nnf .S/ with Ric.!/ D !WP. Given anycompact set K MnS there is a constant CK such that the sectional curvature of Q!tsatisfies

supK

jSec. Q!t /j CK (1.2)

for all small t > 0. Furthermore, on each torus fiber My with y 2 Nnf .S/ we haveQ!t jMyt

! !SF;y ; (1.3)

where !SF;y is the unique flat metric on My cohomologous to !M jMy and where theconvergence is smooth and uniform as y varies on a compact subset of Nnf .S/.

As remarked earlier, in the case of elliptically fibered K3 surfaces (n D 2;m D 1)this theorem follows from the work of Gross and Wilson in [17]. In higher dimensions,in the very special case when S is empty, the theorem (except (1.2)) also follows fromthe work of Fine [11].

The curvature bound (1.2) in Theorem 1.1 does not hold if the generic fibers arenot tori, as one can see for example by taking the product of two nonflat CalabiYaumanifolds with the product Ricci-flat Khler metric and then scaling one factor tozero. On the other hand, we believe that the assumption in Theorem 1.1 that M isprojective is just technical.

We now describe the second main result of this paper, which concerns theGromovHausdorff limit of our manifolds. The GromovHausdorff distance dGH was


introduced by Gromov in the 1980s in [15], and it defines a topology on the space ofisometry classes of all compact metric spaces. For two compact metric spaces X andY , the GromovHausdorff distance of X and Y is

dGH.X;Y / D infZ

dZH .X;Y /

X;Y ,! Z isometric embeddings;

where Z is a metric space and dZH .X;Y / denotes the standard Hausdorff distancebetween X and Y regarded as subsets in Z by the isometric embeddings (see, e.g.,[15], [27] for more background). We would like to understand the GromovHausdorffconvergence of .M; Q!t / in Theorem 1.1. Since the volume of the whole manifoldgoes to zero, the manifolds .M; Q!t / are collapsing. Furthermore, from Theorem 1.1we know that on a Zariski open set of M the Ricci-flat metrics collapse with locallybounded curvature. The collapsing of Riemannian manifolds in the GromovHausdorffsense has been extensively studied from different viewpoints (see, e.g., [2], [5][8], [12], [17], [26], [27], [32] and the references therein). The metric structure ofcollapsed limits of Riemannian manifolds with a uniform lower bound on the Riccicurvature was studied by Cheeger and Colding in [5] and by others. Regarding thecollapsed GromovHausdorff limit of the Ricci-flat metrics in Theorem 1.1, we havethe following result.

First of all, thanks to [35], [46] we know that the diameter of .M; Q!t / satisfiesdiam Q!t .M/ D; for some constant D and for all t > 0. Furthermore, since Q!t !f ! and since the base N is not a point, we also have diam Q!t .M/ D1: Givenany sequence tk ! 0, Gromovs precompactness theorem shows that a subsequenceof .M; Q!tk / converges to some compact path metric space .X;dX / in the GromovHausdorff topology. Note that because of the upper and lower bounds for the diameter,if we rescaled the metrics Q!tk to have diameter equal to 1, the GromovHausdorfflimit (modulo subsequences) would be isometric to .X;dX / after a rescaling.

THEOREM 1.2In the same setting as Theorem 1.1, for any such limit space .X;dX / there is an opendense subset X0 X such that .X0; dX / is locally isometric to .Nnf .S/;!/thatis, there is a homeomorphism W Nnf .S/ ! X0 satisfying, for any y 2 Nnf .S/,that there is a neighborhood By Nnf .S/ of y such that, for y1 and y2 2 By ,

d!.y1; y2/ D dX.y1/; .y2/

:

In fact, we prove that XnX0 has measure zero with respect to the renormalizedlimit measure of [5], which implies that X0 is dense in X . It would be interestingto prove that the metric completion of .Nnf .S/;!/ is isometric to .X;dX /. For K3surfaces, this was proved by Gross and Wilson in [17].


As an application of Theorems 1.1 and 1.2, we study the metric degenerationsof families of polarized hyperkhler manifolds in the large complex structure limit.In [34], Stominger, Yau, and Zaslow proposed a conjecture, known as the SYZ con-jecture, about constructing the mirror manifold of a given CalabiYau manifold viaspecial Lagrangian fibrations. Later another version of the SYZ conjecture was pro-posed by Gross and Wilson in [17], by Kontsevich and Soibelman in [24], and byTodorov via degenerations of Ricci-flat KhlerEinstein metrics. The conjecture saysthat if Mt, t 2 n0 C, is a family of polarized CalabiYau n-manifolds whosecomplex structures tend to a large complex structure limit point when t ! 0, and if!t is the Ricci-flat KhlerEinstein metric representing the polarization on Mt , thenafter rescaling .Mt ;!t / to have diameter 1, they collapse to a compact metric space.X;dX / in the GromovHausdorff sense. Furthermore, a dense open subset X0 Xis a smooth manifold of real dimension n, and the codimension of XnX0 is bigger orequal to 2. This conjecture holds trivially for tori, and was verified for K3 surfacesby Gross and Wilson in [17].

In the third main result of this paper, we consider this conjecture for higher-dimensional hyperkhler manifolds. Let .M;I / be a compact hyperkhler manifoldof complex dimension 2n with a Ricci-flat Khler metric !I . We assume that there isan ample line bundle over M with the first Chern class !I , that we have a holomor-phic fibration f W M ! N as before with N a projective variety, and that there is aholomorphic section s W N ! M . Under these assumptions, it is known that N DCPn(see [21]) and that the smooth fibers of f are complex Lagrangian tori (see [25]). Ifwe perform a hyperkhler rotation of the complex structure, the fibers become specialLagrangian, and we are exactly in the setup of Strominger, Yau, and Zaslow in [34].We furthermore assume that the polarization induced on the torus fibers is principal.In this case, the SYZ mirror symmetry picture predicts that M is a mirror to itself andthat a large complex structure limit is a mirror to a large Khler structure limit. We usethis as our definition of large complex structure limit, so we have a family of polar-ized hyperkhler structures .M; Ls/ with Ricci-flat Khler metric L! which approacha large complex structure limit as s ! 1. By assuming the validity of a standardconjecture on hyperkhler manifolds (Conjecture 2.3), we prove the following.

THEOREM 1.3In the above situation, denote LMs the hyperkhler manifold with period Ls , and letds D diam L!. LMs/. Then, for any sequence sk ! 1, a subsequence of . LMsk ; d2sk L!/converges in the GromovHausdorff sense to a compact metric space .X;dX /, whichhas an open dense subset .X0; dX / locally isometric to an open noncomplete smoothRiemannian manifold .N0; g/ with dimRN0 D .1=2/dimRM .


This proves the conjectures of Gross and Wilson [17], of Kontsevich and Soibel-man [24, Conjecture 1], and of Todorov in our situation, modulo these assumptions,except for the statement that codimR.XnX0/ 2. Again, this was proved by Grossand Wilson in [17, Conjecture 6.2] in the case of K3 surfaces.

This article is organized in the following fashion. In Section 2, we study SYZmirrors of some hyperkhler manifolds, and we derive Theorem 1.3 as a consequenceof Theorems 1.1 and 1.2. In Section 3, we construct semiflat background metrics onthe total space of a holomorphic torus fibration. Theorem 1.1 is proved in Section 4while Theorem 1.2 is proved in Section 5.

2. Hyperkhler mirror symmetryIn this section, we discuss a version of mirror symmetry for hyperkhler manifoldsanalogous to the one used for K3 surfaces in [17]. The situation for general hyper-khler manifolds is considerably less developed, however, and we will make manyassumptions in this discussion. The goal is to explain the background of Theorem 1.3and to show that it follows from Theorems 1.1 and 1.2. This discussion largely repre-sents a summary of known results, and it is completely analogous to [17].

First we review known facts about periods of hyperkhler manifolds. Fix M amanifold of real dimension 4n which supports a hyperkhler manifold structure withholonomy being the full group Sp.n/. Set L D H 2.M;Z/, set LR WD LZR, and setLC WD L Z C. Then there is a real-valued nondegenerate quadratic form qM W L !R, called the BeauvilleBogomolov form, with the property that there is a constant csuch that

qM ./n D c

ZM

2n

for 2 L, of signature .C;C;C;; ;/. We write qM .; / for the induced pairing,with qM .;/ D qM ./.

We can define the period domain of M to be

PM WD 2 P.LC/

qM ./ D 0; qM .; N/ > 0

:

The Teichmller space of M , TeichM , is the set of hyperkhler complex structures onM modulo elements of Diff0.M/, the diffeomorphisms of M isotopic to the identity.By the BogomolovTianTodorov theorem, this is a (non-Hausdorff) manifold. Thereis a period map

Per W TeichM ! PMtaking a complex structure on M to the class of the line H 2;0.M/. Then Per is tale,and was proved to be surjective by Huybrechts in [20]. While recently Verbitsky [40]


proved a suitably formulated global Torelli theorem, one must keep in mind that Peris not, in general, a diffeomorphism.

Next consider a complex structure on M and Ricci-flat Khler metric !I mak-ing M hyperkhler. Then a choice of a holomorphic symplectic 2-form I , alongwith !I , completely determines this structure. In particular, if we write I D !J Cp1!K , we can normalize I so that qM .!I / D qM .!J / D qM .!K/. Furthermore,necessarily qM .!I ;!J / D qM .!I ;!K/ D qM .!J ;!K/ D 0. The triple !I ;!J ;!Kis called a hyperkhler triple. It gives rise to an S2-worth of complex structurescompatible with the same hyperkhler metric: in particular, one has the J com-plex structure with holomorphic symplectic form J WD !K C

p1!I and Khlerform !J , and the K complex structure with holomorphic symplectic form K WD!I C

p1!J and Khler form !K .Now suppose that we are given a complex structure on M such that there is

a fibration f W M ! N , with fibers being holomorphic Lagrangian subvarieties ofM . Suppose furthermore that N is a Khler manifold. Then by results of Matsushita(see [25] and [16, Proposition 24.8]), the smooth fibers of f are complex tori andN is a Fano manifold with b2.N / D 1. Furthermore, if M is projective of complexdimension 2n, then N D CPn by a result of Hwang [21, Theorem 1.2]. If My is afiber of f , then !J jMy D !K jMy D 0, from which it follows that Im.nK/jMy D 0,so that after hyperkhler rotation the fibers of f are special Lagrangian.

The StromingerYauZaslow conjecture in [34, p. 244] predicts that mirror sym-metry can be explained via dualizing such a special Lagrangian torus fibration. In ageneral situation, it can be hard to dualize torus fibrations, because of singular fibers.The case that M is a K3 surface, treated in detail in [17], is rather special becausePoincar duality gives a canonical isomorphism between a 2-torus and its dual.

With some additional assumptions, a similar situation holds in the hyperkhlercase. Suppose that the Khler form !I is integral so that there is an ample line bundleL on X whose first Chern class is represented by !I . The restriction of this line bun-dle to a nonsingular fiber My then induces a polarization of some type .d1; : : : ; dn/.In particular, there is a canonical map My ! M_y given by

My 3 x 7! LjMy txL1jMy 2 M_y :

Here M_y is the dual abelian variety to My , classifying degree zero line bundles onMy , and tx W My ! My is given by translation by x, which makes sense once onechooses an origin in My . The kernel of this map is .Z=d1Z Z=dnZ/2. Inparticular, if f possesses a section s W N ! M , and if N0 WD N n f .S/ where S isas in the introduction, then the dual of f 1.N0/ ! N0 can be described as a quotientmap, given by dividing out by the kernel of the polarization on each fiber. One canthen hope that this dual fibration can be compactified to a hyperkhler manifold.


In general, if My carries a polarization of type .d1; : : : ; dn/, it is not difficult tocheck that the dual abelian variety M_y carries a polarization of type .dn=dn; dn=dn1;: : : ; dn=d1/. Thus if the polarization is not principal, the SYZ dual manifold need notcoincide with M (see examples of nonprincipally polarized fibrations in [30, Exam-ple 3.8, Remark 3.9]). It is possible that Sawons fibrations do not have duals whichare hyperkhler manifolds, as a natural compactification might be a holomorphic sym-plectic variety without a holomorphic symplectic resolution of singularities.

On the other hand, if !I induces a principal polarization on each fiber My(i.e., the map My ! M_y is an isomorphism), then the SYZ dual of the fibrationf 1.N0/ ! N0, assuming again the existence of a section, can be canonically iden-tified with f 1.N0/ ! N0, and thus it is natural to consider f W M ! N to be aself-dual fibration, at least at the purely topological level. In this case, and only inthis case, SYZ mirror symmetry predicts that hyperkhler manifolds are self-mirror.The idea that hyperkhler manifolds should be self-mirror was first suggested andexplored by Verbitsky in [38].

In this case only, we can be more explicit about mirror symmetry. We summarizeour assumptions so far with the following.

Assumptions 2.1Let MI be a hyperkhler manifold with f W MI ! N a complex torus fibration, alongwith a section s W N ! MI and an ample line bundle L with first Chern class repre-sented by a hyperkhler metric !I . We assume further that the induced polarizationon the smooth fibers of f is principal and that N is projective.

Thus, with these assumptions, it is natural to assume that mirror symmetryexchanges complex and Khler moduli for the fixed underlying space M . This canbe described at the level of period domains as follows.

Let 2 LR be the class represented by !I . Fix an integral Khler class !N onN , and let E 2 L be represented by f !N , so that qM .E/ D 0.

LEMMA 2.2In the above situation, we have qM .E;/ 6D 0.

ProofBy [16, Exercise 23.2], we have

qM .E;/

ZM

2n D 2qM ./ZM

2n1 ^ f !N 6D 0;

so qM .E;/ 6D 0.


Denote by E? LR the orthogonal complement of E under qM , and denote byE?=E the quotient space E?=RE . Then qM induces a quadratic form on E?=E .Let

C.M/ WD x 2 E?=E qM .x/ > 0;and define the complexified Khler moduli space of M to be

K.M/ WD .E?=E/ iC.M/ .E?=E/C:

We then have an isomorphism

mE; W K.M/ ! PM nE?

via, representing an element of .E?=E/C by 2 E? C,

7!h 1qM .E;/

C 12

qM ./qM .E;/2

C qM ./C 2 qM .; /qM .E;/

E

i:

Indeed, one first checks to verify that this is independent of whichever representa-tive might be chosen. One then notes that the coefficient of E is chosen so thatqM .mE; .// D 0, and qM .mE; ./;mE; . N // D 2qM .Im/ > 0 by assumptionthat 2 K.M/. Further, mE; is clearly injective, since D mE; ./=qM .E;/modE . It is surjective, since given 2 PM n E?, we can rescale so thatqM .;E/ D 1, and then D mE; . =qM .E;/modE/.

We can then view the mirror map mE; described above as realizing mirror sym-metry on the level of period domains, defining an exchange of data

.M;;B C p1!/ $ .M; L; LB C p1 L!/:

Here ; L 2 PM , with qM .E;/; qM .E; L/ 6D 0, so that we can assume that and L are normalized with qM .E;/ D qM .E; L/ D 1. Furthermore, B; LB 2 E?=Eand !; L! 2 E? satisfy qM .!;/ D qM . L!; L/ D 0 and qM .!/; qM . L!/ > 0. Therelationship between the two triples is that L D mE; .B C

p1!/ and LB; L! arethe unique cohomology classes satisfying the above conditions and D mE; . LB Cp1 L!/. Indeed, LB and L! exist, since as qM .E;/ D 1, we can write D.1=qM .E;// C LB C

p1 L! modE , and replacing a chosen representative L! withL! .qM . L!;/=qM .E;/ qM . L!;B//E , one guarantees that qM . L; L!/ D 0.

This mirror symmetry on the level of period domains does not quite give an exactmirror symmetry on the level of moduli spaces, since global Torelli does not in generalhold, so there might be a number of choices of complex structure on M with period. In addition, ! or L! need not represent a Khler form except for very generalchoices of complex structure.


Nevertheless, this allows us to identify a large complex structure limit as beingmirror to a large Khler limit. The family

M; D 1qM .E;/

C LB C p1 L! mod E; s!

represents a large Khler limit, with the Khler class moving off to infinity while thecomplex structure is fixed, and this is mirror to the triple

M; Ls D 1qM .E;/

C p1s! mod E; LB C p1 L!:

If, for each s, we have an actual hyperkhler manifold with period Ls and Khlerform L!, then we would like to understand the limiting metric behavior.

To do so, we use hyperkhler rotation, and to do this we need to normalize theholomorphic symplectic form, defining

Lnors D s1s

qM . L!/qM .!/

Ls :

Then we have qM .Re Lnors / D qM .Im Lnors / D qM . L!/. So Re Lnors , Im Lnors , and L!form a hyperkhler triple, and hence we can hyperkhler rotate to obtain a hyperkhlermanifold with holomorphic 2-form

Ls;J WD Im Lnors Cp1 L! D

sqM . L!/qM .!/

! qM .!;/

qM .E;/E

C p1 L!

and Khler form

L!s;J D Re Lnors Ds

qM . L!/qM .!/

h1s

1qM .E;/

12

qM ./

qM .E;/2E

C s

2qM .!/E

i:

We note that the period Ls;J is in fact independent of s, so we can fix the complexstructure on M independent of s. Assume that E is the first Chern class of a nefline bundle on M with respect to a complex structure with period Ls;J , and assumethat L!s;J is a Khler class with respect to this complex structure if s s0, for somes0 0. We now take s D s0

p.t C 1/=t , so that as t goes to zero, s goes to infinity

and we define the rescaled metrics

L!nort;J Dp

t .t C 1/ L!s.t/;J D t L!s0;J Cs0

2

pqM . L!/qM .!/E:

So as t ! 0, L!nort;J moves on a straight line towards .s0=2/p

qM . L!/qM .!/E , andL!s0;J is Khler.

To relate this to the results of this paper, we have the following conjecture, statedin [18], [39].


Conjecture 2.3Let M be an irreducible hyperkhler manifold, and let L be a nontrivial nef bundleon M , with qM .c1.L// D 0. Then L induces a holomorphic map f 0 W M ! N 0 to aprojective variety N 0 with Lm f 0.O.1// for some m> 0.

If such a map exists, it is necessarily a holomorphic Lagrangian fibration. If fur-thermore M is projective, then N 0 D CPn by [21]. This conjecture follows from thelog abundance conjecture if some multiple of L is effective, and it has been studied,for example, in [1], [4], [18], and [39].

Let us suppose this conjecture holds. By choosing s0 properly, we assume that.s0=2/

pqM . L!/qM .!/ is an integer and thus .s0=2/

pqM . L!/qM .!/E D f 0 for

an ample class on N 0, where f 0 and N 0 are obtained by Conjecture 2.3. Becauseof the hyperkhler rotation, the Riemannian metrics defined by . Lnors ; L!/ and by. Ls;J ; L!s;J / are the same. Therefore, to understand the GromovHausdorff limit ofthe large complex structure limit .M; Lnors ; L!/ (this is the same one that appears inthe statement of Theorem 1.3), we can instead consider .M; Ls;J ; L!s;J /. Now Ls;Jis independent of s, so we are simply changing the Khler class, and the rescaled met-rics L!t;J D

pt .t C 1/ L!s.t/;J D t L!s0;J C f 0 move towards f 0 along a straight

line. Therefore, we are exactly in the setting of Theorems 1.1 and 1.2, which describethe GromovHausdorff limit of .M; L!t;J / as t goes to zero. But as noted in the intro-duction, we also have that the diameter of L!t;J is bounded uniformly away from zeroand infinity, so if we further rescale the metrics L!t;J to have diameter 1, then up toa subsequence the GromovHausdorff limit only changes by a rescaling, and Theo-rem 1.3 follows.

3. Semiflat metricsIn this section, we discuss semiflat forms and metrics, extending some results from[17] and [19] to our setting.

In general, a closed real .1; 1/-form !SF on an open set U MnS will be calledsemiflat if its restriction to each torus fiber My \ U with y 2 f .U / is a flat metric,which we will always assume to be cohomologous to !M jMy . If !SF is also Khler,then we will call it a semiflat metric. Semiflat forms can also be defined when thefibers My are not tori but general CalabiYau manifolds, by requiring that the restric-tion to each fiber be Ricci-flat (see [33], [36]). These forms were first introduced byGreene, Shapere, Vafa, and Yau in [14].

Fix now a small ball B Nnf .S/ with coordinates y D .y1; : : : ; ym/, and con-sider the preimage f W U D f 1.B/ ! B . This is a holomorphic family of com-plex tori, and if B is small enough it has a holomorphic section 0, which we alsofix. We can then define a complex Lie group structure on each fiber My D f 1.y/


with unit 0.y/. We claim that this family is locally isomorphic to a family of theform f 0 W .B Cnm/= ! B; where h W ! B is a lattice bundle with fiberh1.y/ D y Z2n2m, so that My Cnm=y : To see this, note that each fiberMy D f 1.y/ is a torus biholomorphic to Cnm=y for some lattice y that variesholomorphically in y. We choose a basis v1.y/; : : : ; v2n2m.y/ of this lattice, whichvaries holomorphically in y. Given these lattices, we can construct the family f 0 bytaking the quotient of B Cnm by the Z2n2m-action given by .n1; : : : ; n2n2m/ .y; z/ D y; z C Pi nivi .y/, where z D .z1; : : : ; znm/ 2Cnm. Note that differentchoices of generators give isomorphic quotients. By construction, the fiber f 01.y/ isbiholomorphic to f 1.y/ for all y 2 B . A theorem of Kodaira and Spencer [22, The-orem 14.3] (see also [41, Satz 3.6]) then implies that the families f and f 0 are locallyisomorphic, so up to shrinking B there exists a biholomorphism .B Cnm/= ! Ucompatible with the projections to B , proving our claim. With this identification, thesection 0 W B ! U is induced by the map B ! B Cnm given by y 7! .y; 0/.

Composing this biholomorphism with the quotient map B Cnm ! .B Cnm/= by the Z2n2m-action, we get a holomorphic map p W B Cnm ! Usuch that f p.y; z/ D y for all .y; z/, and p is a local isomorphism (the map p isalso the universal covering map of U ).

We now assume that M is projective and that !M is an integral class, so eachcomplex torus fiber My , y 2 B , can be polarized by !M , which gives an amplepolarization of type .d1; : : : ; dnm/ for some sequence of integers d1jd2j jdnm.By [3, Proposition 8.1.1], one can then assume that is generated by d1e1; : : : ;dnmenm;Z1; : : : ;Znm 2 Cnm, where e1; : : : ; enm is the standard basis forCnm. Furthermore, the matrix Z with columns Z1; : : : ;Znm must satisfy Z D Ztand ImZ positive definite. Also, on the fiber My , the Khler formP

i;j

p1.ImZ/ij dzi ^ d Nzj is cohomologous to !M jMy . Let

gij D .ImZ/1ij :

Note that Z depends on y 2 B , as does gij . Recall that we have the fiber coordi-nates z1; : : : ; znm. Consider the function

.y; z/ DXi;j

gij .y/2

.zi Nzi /.zj Nzj /

:

We would first like to show thatp1@N@ is invariant under translation by flat

sections of the GaussManin connection on B Cnm (this is the connection on thisbundle such that sections of are flat sections of the bundle). It is enough to checkinvariance under translation by s for s one of the generators of , 2 R. First,consider the composition of with a general translation zi 7! zi C i .y/:

COLLAPSING OF ABELIAN FIBERED CALABIYAU MANIFOLDS 529Xi;j

gij2

.zi C i Nzi Ni /.zj C j Nzj Nj /

D

Xi;j

gij

2

.i Ni /.zj Nzj /C .j Nj /.zi Nzi /C .i Ni /.j Nj /

D

Xi;j

gij

.i Ni /.zj Nzj /C 1

2.i Ni /.j Nj /

;

the last equality by the symmetry gij D gj i . We now consider two cases. If i D ikfor some k, so that i is real, then in fact the above formula reduces to , so is itselfinvariant under this translation. Secondly, if we take i D Zik for some k, 2 R,we obtain

Xi;j

.ImZ/1ij2

p1.ImZ/ik.zj Nzj / 22.ImZ/ik.ImZ/jk

D Xj

2jkp1.zj Nzj / 22jk.ImZ/jk :

Applying @N@ kills the correction term, so p1@N@ is invariant under this action. Thismeans that

p1@N@ is the pullback under p of a 2-form !SF on Up!SF D

p1@N@; (3.1)and that !SF is semiflat since its restriction to a fiber is

p1Pi;j gij .y/dzi ^ d Nzj ,a flat metric on My cohomologous to !M jMy . Note that the function on B Cnmhas the scaling property

.y;z/ D 2.y; z/; (3.2)for all 2R.

We now claim that on U the semiflat form !SF is positive semidefinite. Tocheck this, it is enough to check at one point on each fiber, because of the invari-ance of this form. We check at the point z1 D D znm D 0, where the formis

p1Pi;j gij dzi ^ d Nzj , which is clearly positive semidefinite. It follows that!SF 0, and moreover that given any Khler metric !0 on B the form !SF C f !0is a semiflat Khler metric on U .

Suppose now that we have a holomorphic section W B ! U of the map f .We will denote by T W U ! U the fiberwise translation by (with respect to thesection 0). If we choose any local lift of to B Cnm, given by y 7! .y; Q.y//,then the translation T is induced by the map B Cnm ! B Cnm given by.y; z/ 7! .y; z C Q.y// (the choice of lift Q is irrelevant). We also have a map T WU ! U given by fiberwise translation by (with respect to 0), which is inducedby .y; z/ 7! .y; z Q.y//. The two translations are biholomorphisms of U and are


inverses to each other. For later purposes, we will need the following version of the@N@-Lemma, which is analogous to [17, Lemma 4.3] (see also [19, Proposition 3.7]),except that we work away from the singular fibers.

PROPOSITION 3.1Let ! be any Khler metric on U cohomologous to !SF in H 2.U;R/. Then thereexist a holomorphic section W B ! U of f and a smooth real function on U suchthat

T !SF ! Dp1@N@ (3.3)

on U .

If in addition ! is also semiflat, then is constant on each fiber My and istherefore the pullback of a function from B .

ProofBy assumption there is a 1-form on U such that

!SF ! D d D @0;1 C N@1;0; N@0;1 D 0;where D 0;1 C 1;0 and 0;1 D 1;0.

We claim that .0; 1/-forms

j Dp1 N@

nmXiD1

gij .y/.zi Nzi /; j D 1; : : : ; nm; (3.4)

are invariant under translations by flat sections of the GaussManin connection onB Cnm, and thus descend to .0; 1/-forms on U . It is enough to check invarianceunder translation by s where s is a generator of and where 2R. First, considera general translation zi 7! zi C i .y/. If i D ik for some k, so that i is real, then

j are invariant. If i D Zik for some k, 2R, we obtain

nmXiD1

gij .zi C Zik Nzi NZik/

DnmXiD1

gij .zi Nzi /

C 2p1nmXiD1

.ImZ/1ij .ImZ/ik

DnmXiD1

gij .zi Nzi /C 2p1jk :


Applying N@ kills the correction term, so j are invariant, and therefore they define.0; 1/-forms on U . Since, for any y 2 B ,

p.j jMy / D p1

nmXiD1

gij .y/d Nzi ; (3.5)

is fiberwise constant and gij is nondegenerate, we have that i jMy , i D 1; : : : ; nmis a basis of H 0;1.My/.

We claim that there are holomorphic functions i W B !C such that

0;1 DnmXiD1

ii C N@h; (3.6)

for a complex-valued functionh onU . To prove this, note thatH 0;1.U / D H 1.U;OU /,which by the Leray spectral sequence for f is isomorphic to H 0.B;R1fOU / sinceH k.B;fOU / D H k.B;OB/ D 0 for k 1. It follows that a N@-closed .0; 1/-form onU represents the zero class if and only if its restriction to My represents the zero classin H 0;1.My/ for all y 2 B . Consider now the .0; 1/-forms d Nyi , 1 i m, on B anddenote their pullbacks to U by the same symbol. Then at each point of U the formsj ; 1 j nm together with d Nyi; 1 i m, form a basis of .0; 1/-forms. Wecan then write

0;1 DnmXjD1

wj j CmX

iD1hi d Nyi ;

where wj ; hi are smooth complex functions on U . If we now restrict to a fiber My weget 0;1jMy D

PnmjD1 wj j jMy ; and the functions wj restricted to My can be thought

of as functions on Cnm which are periodic with period y . There is a holomorphicT 2n2m-action on U which is induced by the action of R2n2m on B Cnm givenby x .y; z/ D y; z C Pj xj j .y/, where j .y/ is a basis for the lattice y (thechoice of which is irrelevant). If is a function or differential form on U or My , wewill denote by Q its average with respect to the T 2n2m-action. In particular, if isa function on U then Q is the pullback of a function from B . We now call j D Qwj ,1 j nm, which are functions of y 2 B only. We clearly have that Qj D j andgd Nyi D d Nyi , so

0;1jMy DnmXjD1

j .y/j jMy :

Now the T 2n2m-action on My is generated by holomorphic vector fields and there-fore acts trivially on the Dolbeault cohomology H 0;1.My/, which implies that


0;1jMy D 0;1jMy DnmXjD1

j .y/j jMy ;

in H 0;1.My/ for all y 2 B . If we show that the j .y/ are holomorphic, then the.0; 1/-form 0;1 Pj j .y/j on U would be N@-closed and cohomologous to zeroin H 0;1.U /, thus proving (3.6).

Now we call Vj , 1 j n m and Wi , 1 i m the T 2n2m-invariant.0; 1/-type vector fields on U which are the dual basis to j ; d Nyi . We have Vj Dp1PnmkD1 gjk @@ Nzk , where gjk is the inverse matrix of gjk and the vector fields @@ Nzkare well defined on U . We will not need the explicit formula for Wi , but just the factthat if a function f on U is the pullback of a function on B , then Wi .f / D @f@ Nyi .

To see why j .y/ is holomorphic, compute

0 D N@0;1 DXi;j

Wi .wj / d Nyi ^ j CXi;j

Vi .wj /i ^ j

CXi;j

Wj .hi / d Nyj ^ d Nyi CXi;j

Vj .hi /j ^ d Nyi :

Since each Vj is a linear combination of @@ Nzk , we have that the functions Vi .wj / andVj .hi / have average zero on each fiber. Taking the average then gives

0 D N@g0;1 D Xi;j

@j

@ Nyi d Nyi ^ j C

Xi;j

@ Qhi@ Nyj d Ny

j ^ d Nyi :

Since the forms d Nyi ^ j and d Nyj ^d Nyi are linearly independent at every point, thisimplies that j .y/ are indeed holomorphic.

Now let T W U ! U be the translation induced by the section D .p 1; : : : ;p nm/, where p W B Cnm ! U is the quotient map. SinceX

i;j

gij2

.zi C i Nzi Ni /.zj C j Nzj Nj /

D

Xi;j

gij

2

.i Ni /.zj Nzj /C .j Nj /.zi Nzi /C .i Ni /.j Nj /

D

Xi;j

gij

.i Ni /.zj Nzj /C 1

2.i Ni /.j Nj /

;

we have

pT !SF p!SF D p1@N@

Xi;j

gij .i Ni /.zj Nzj /Cp1@[email protected]/

D p@

Xi

ii N@X

i

ii

C p1@[email protected]/;


where .y/ D Pi;j .gij =2/.i Ni /.j Nj / is a real function of y only. We havejust proved that

!SF ! D @0;1 C N@0;1 D @X

i

ii C @N@hC N@X

i

ii C N@@ Nh:

Thus

pT !SF p! D pp1@[email protected] ImhC/;

which proves (3.3) with D 2 ImhC.

4. Estimates and smooth convergenceIn this section, we prove a priori estimates of all orders for the Ricci-flat metrics Q!twhich are uniform on compact sets of MnS , and then use these to prove Theorem 1.1.These estimates improve the results in [36], and use crucially the assumptions that Mis projective and that the smooth fibers My are tori.

From now on we fix a small ball B Nnf .S/, and as before we call U Df 1.B/ and we have the holomorphic covering map p W B Cnm ! U , with f p.y; z/ D y where .y; z/ D .y1; : : : ; ym; z1; : : : ; znm/ the standard coordinates onB Cnm.

LEMMA 4.1There is a constant C such that on U the Ricci-flat metrics Q!t satisfy

C1.!0 C t!M / Q!t C.!0 C t!M /; (4.1)

for all small t > 0.

ProofThis estimate is contained in the second-named authors work [36], although it is notexplicitly stated there. To see this, start from [36, (3.24)], which gives a constant Cso that on U we have

C1.t!M / Q!t :

Then use [36, Lemma 3.1] to get

C1!0 Q!t ;

and so adding these two inequalities we get

C1.!0 C t!M / Q!t ;


or in other words tr Q!t!t C on U , where !t D !0 C t!M as before. To get thereverse inequality, we note that on U we have

tr!t Q!t .tr Q!t!t /n1Q!nt!nt

C Q!nt

!nt C;

where the last inequality follows from [36, (3.23)]. We thus get the reverse inequality

Q!t C.!0 C t!M /;

thus proving (4.1).

We now let t W B Cnm ! B Cnm be the dilationt .y; z/ D

y;

zpt

;

which takes the latticep

ty to y . If we pull back the Khler potential 't on Uvia p, we get a function 't p on B Cnm which is periodic in z with period y ,that is, 't p.y; z C `/ D 't p.y; z/ for all ` 2 y . The function 't p t is thenperiodic in z with period

pty . Note that since !0 is the pullback of a metric from

Nnf .S/, we have t p!0 D p!0:Recall now that we have a positive semidefinite semiflat form !SF on U , and

that !0 C !SF is then a semiflat Khler metric on U . Since U is diffeomorphic to aproduct B My , it follows that !SF and !M are cohomologous on U . We now applyProposition 3.1 and get a holomorphic section W B ! U and a real function on Usuch that

T !SF !M Dp1@N@ (4.2)

on U , where T is the fiberwise translation by .

LEMMA 4.2There is a constant C such that on the whole of B Cnm we have

C1p.!0 C!SF / t pT Q!t Cp.!0 C!SF / (4.3)


ProofFirst of all notice that after replacing U with a slightly smaller open set, the semiflatmetric !0 C!SF is uniformly equivalent to !M , which implies that

C1.!0 C t!SF / !0 C t!M C.!0 C t!SF / (4.4)


for all small t > 0. Thanks to Lemma 4.1 on U we have that

C1.!0 C tT !M / T Q!t C.!0 C tT !M /;and since T !M is uniformly equivalent to !M we also have that

C1.!0 C t!M / T Q!t C.!0 C t!M /; (4.5)and combining (4.4) and (4.5) we get

C1.!0 C t!SF / T Q!t C.!0 C t!SF /; (4.6)on U . If we pull back (4.6) by p t we get

C1.p!0 C tt p!SF / t pT Q!t C.p!0 C tt p!SF /; (4.7)on all of B Cnm. We claim that on the whole of B Cnm we have that

tt p!SF D p!SF : (4.8)In fact, the construction of !SF in Section 3 gives that p!SF D

p1@N@; for afunction on B Cnm that satisfies

t .y; z/ D y;

zpt

D 1

t.y; z/; (4.9)

for all .y; z/ in B Cnm and any t > 0. It follows then thattt p!SF D tt

p1@N@ D tp1@N@. t / Dp1@N@ D p!SF ; (4.10)

as claimed. Combining (4.7) and (4.8) we get the bound (4.3).

PROPOSITION 4.3Given any compact set K in B Cnm and any k 0 there exists a constant Cindependent of t > 0 such that

kt pT Q!tkCk.K;/ C; (4.11)where is the Euclidean metric on B Cnm.

ProofWe pull back (1.1) via T p t and get

.t pT Q!t /n.y; z/ D ct tnm.t pT !M /n.y; z/D ct .pT !M /n

y;

zpt

;


since the pullback under t of any volume form f .y; z/dy1 ^ ^ d Nznm on B Cnm equals tmnf .y; z=

pt / dy1 ^ ^d Nznm: We now claim that in fact we have

.pT !M /ny;

zpt

D .pT !M /n.y; z/:

To see this, consider the .n; 0/-form

dy1 ^ ^ dym ^ dz1 ^ ^ dznm

on B Cnm. This form is invariant under the Z2n2m-action described above

.n1; : : : ; n2n2m/ .y; z/ Dy; z C

Xi

nivi .y/;

where .y; z/ D .y1; : : : ; ym; z1; : : : ; znm/, and so it descends to a holomorphic .n; 0/-form to the quotient .B Cnm/=, and using the biholomorphism with U we get aholomorphic .n; 0/-form on U . We can then consider the volume form .

p1/n2^, and we have

T !nM D h .p1/n2^;

where h is a smooth positive function on U . Takingp1@N@ log of both sides we get

p1@N@ logh D p1@N@ log T!nM

.p1/n2^ D 0;

since T !M is Ricci-flat and is a holomorphic .n; 0/-form. So logh is pluri-harmonic on U , and this implies that its restriction to any fiber My with y 2 B isconstant. Pulling back via p we get

.pT !nM /.y; z/ D .h p/.y; z/.p1/n2 dy1 ^ ^ d Nznm;

but since h is constant along the fibers of f and p is compatible with the projectionto B we get that the function .h p/.y; z/ on B Cnm is independent of z. Inparticular we have

.pT !M /ny;

zpt

D .pT !M /n.y; z/;

and so the rescaled metrics t pT Q!t satisfy the nondegenerate complex MongeAmpre equation

.t pT Q!t /n D .p!0 C tt pT !M Cp1@N@ Q't /n D ct .pT !M /n

on B Cnm, where we have set


Q't D 't T p t :We claim that the estimates (4.11) hold. To see this, we use (4.2) and get

p!SF D pT !M C pT p1@N@ (4.12)

for a function on U . On B Cnm we can then use (4.10) and (4.12) and writet pT Q!t D p!0 C tt pT !M C

p1@N@ Q'tD p!0 C tt p.!SF T

p1@N@/C p1@N@ Q'tD p!0 C p!SF tt pT

p1@N@ C p1@N@ Q'tD p.!0 C!SF /C

p1@N@ut ; (4.13)where for simplicity we write ut D Q't t . T p t /: The functions ut areuniformly bounded in C 0.B Cnm/ because of the L1 bound for 't from [9], [10]and because is a fixed function on U . The functions ut satisfy the complex MongeAmpre equations

.p!0 C p!SF Cp1@N@ut /n D ct .pT !M /n (4.14)

on B Cnm, and on any compact subset K of B Cnm the Khler metric p.!0 C!SF / is C1 equivalent to the Euclidean metric (with constants that depend only onK). The bounds (4.3) imply that

C1 p.!0 C!SF /Cp1@N@ut C

on K for all small t > 0, where C depends on K . The constants ct are boundeduniformly and away from zero, because by definition we have

limt!0 ct D

n

m

RM

!m0 ^!nmMRM

!nM> 0 (4.15)

(see also [10], [36, (2.6)]). After shrinking K slightly we can then apply the EvansKrylov theory (as explained, e.g., in [13], [31]) and Schauder estimates to get higher-order estimates kutkCk.K;/ C.k/ for all k 0, thus proving (4.11).

LEMMA 4.4Given any compact set K MnS there is a constant CK such that the sectionalcurvature of Q!t satisfies

supK

jSec. Q!t /j CK (4.16)



ProofWe can assume that K is sufficiently small so that f .K/ B for a ball B as before,and that there is a compact set K 0 B Cnm so that p W K 0 ! T .K/ is a biholo-morphism. We then have

supK

jSec. Q!t /j D supT .K/

jSec.T Q!t /j D supK0

jSec.pT Q!t /j

D sup1t .K

0/

jSec.t pT Q!t /j:

For t > 0 small enough, the sets 1t .K/ are all contained in a fixed compact setK 00 B Cnm. From (4.3) and (4.11) we then get a uniform bound for the sectionalcurvatures of t pT Q!t on K 00, and this proves (4.16).

LEMMA 4.5Given any compact set K in B Cnm and any k 0 there exists a constant Cindependent of t > 0 such that

kpT Q!tkCk.K;/ C; (4.17)where is the Euclidean metric on B Cnm.

ProofGiven K for all t > 0 small enough the sets 1t .K/ are all contained in a fixedcompact set K 0 B Cnm. We wish to deduce (4.17) from (4.11). To see this,write on B Cnm

t pT Q!t Dp1

Xi;j

Ai Nj .t; y; z/dzi ^ d Nzj C

Xi;j

Bi Nj .t; y; z/dyi ^ d Nyj

CXi;j

Ci Nj .t; y; z/dyi ^ d Nzj C

Xi;j

Di Nj .t; y; z/dzi ^ d Nyj

:

Thanks to (4.11), on K 0 the coefficients A;B;C;D satisfy uniform C k-estimates inthe variables .y; z/ independent of t . We then pull back this equation via the map 1=t(the inverse of t ) and get

pT Q!t Dp1

tXi;j

Ai Nj .t; y; zp

t / dzi ^ d Nzj CXi;j

Bi Nj .t; y; zp

t / dyi ^ d Nyj

C ptXi;j

Ci Nj .t; y; zp

t / dyi ^ d Nzj

C ptXi;j

Di Nj .t; y; zp

t / dzi ^ d Nyj;


and the new coefficients are uniformly bounded in C k on K , thus proving (4.17).

PROPOSITION 4.6As t goes to zero, we have

Q!t ! f !

in C1loc .MnS;!M /, where ! D !N Cp1@N@' is a Khler metric on Nnf .S/ with

Ric.!/ D !WP as in Theorem 1.1.

ProofRecall that Q!t D !0 C t!M C

p1@N@'t , so that

pT Q!t D p!0 C tpT !M Cp1@N@.'t T p/:

We now fix a compact set K MnS , which we can assume is sufficiently small sothat f .K/ B for a ball B as before, and that there is a compact set K 0 B Cnmsuch that p W K 0 ! T .K/ is a biholomorphism. From (4.17) (together with the L1bound for 't from [9], [10]) we see that

k't T pkCk.K0;/ C.k/;

and therefore also

k'tkCk.K;!M / C.k/; (4.18)

since T p W K 0 ! K is a fixed biholomorphism. From [36] we know that 't !f ' in C 1;loc .MnS;!M /, and so (4.18) implies that 't ! f ' in C1loc .MnS;!M /,and therefore that Q!t ! f ! in C1loc .MnS;!M /.

As a corollary of this, for any compact subset K MnS , there is a positivefunction ".t/ which goes to zero as t ! 0, such that

f ! ".t/!M Q!t f ! C ".t/!M (4.19)

on K , as well as

e".t/f ! Q!t : (4.20)

We now finish the proof of Theorem 1.1. We have already proved the first twostatements in Proposition 4.6 and Lemma 4.4, and it remains to prove (1.3). First, weneed the following lemma.


LEMMA 4.7As t goes to zero, we have

t pT Q!t ! p.!SF C f !/ (4.21)

in C1loc .B Cnm; /, where is the Euclidean metric.

ProofRecall that from (4.13) we see that on B Cnm

t pT Q!t D p.!0 C!SF /Cp1@N@ut ;

where the functions ut D Q't tt pT have uniform C1-bounds on compactsets. We need to show that as t goes to zero we have ut ! .f p/' in C1loc .B Cnm; /, where f ' is the C 1;-limit of 't from [36]. To prove this we need anotherestimate from the second authors work [36, (3.9)], which implies that there is a con-stant C (that depends on the initial choice of B) so that, for all 0 < t 1, we have

supy2B

oscMy't Ct: (4.22)

We now use this together with the fact that 't ! f ' in C 0 to get that, for any .y; z/in B Cnm, we have

j Q't .y; z/ .f p/'.y; z/j D't T p

y;

zpt

'.y/

't p

y;

zpt Q.y/

't p.y; z/

C 't p.y; z/ .f '/ p.y/

Ct C supU

j't f 'j;

where in the last line we used (4.22) because the points p.y; .z=pt / Q.y// andp.y; z/ lie in the same fiber My . Letting t go to zero we see that Q't ! .f p/' inC 0.B Cnm/. On the other hand we have that tt p ! 0 in C 0.B Cnm/, andso ut ! .f p/' in C 0.BCnm/. Thanks to the higher-order estimates for ut , wealso have that ut ! .f p/' in C1loc .B Cnm; /, up to shrinking B slightly.

We can now complete the proof of Theorem 1.1.

ProofRecall that thanks to Lemma 4.7, on B Cnm we can write


t pT Q!t p.!SF C f !/ D Et ;where the error term Et is a .1; 1/-form that goes to zero smoothly on compact sets.From (4.8) we also have that

Et D t p.T Q!t f ! t!SF /:If we restrict the form T Q!t f ! t!SF to a fiber My and divide by t , we get

Et

t

yCnmD

t p

T Q!t jMyt

!SF;y:

Pulling back this via the map 1=t (the inverse of t ) we get

1=tEt

t

yCnmD p

T Q!t jMyt

!SF;y:

Explicitly we have 1=t .y; z/ D .y; zp

t /, which implies that 1=t

dzi D pt dzi , andso

1=t

Et

t

yCnm .y; z/ D Et jyCnm.y; z

pt /;

which goes to zero smoothly as t approaches zero, uniformly in y. It follows that.T Q!t jMy=t/ converges smoothly to !SF;y , and the convergence is uniform as yvaries on compact sets of Nnf .S/. Pulling back via T , and using the fact thatT !SF;y D !SF;y , we see that . Q!t jMy=t/ also converges smoothly to !SF;y , asdesired.

Remark 4.8Note that in particular we get the estimate

supMy

jr. Q!jMy /j2!M Ct2;

which improves [36, (2.11)].

5. GromovHausdorff convergenceIn this section, we study the collapsed GromovHausdorff limits of the Ricci-flatmetrics Q!t and prove Theorem 1.2.

LEMMA 5.1There is an open subset X0 X such that .X0; dX / is locally isometric to .N0;!/where N0 D Nnf .S/, that is, there is a homeomorphism W N0 ! X0 such that, forany y 2 N0, there is a neighborhood By N0 of y satisfying that, if y1 and y2 2 By ,


d!.y1; y2/ D dX.y1/; .y2/

:

Furthermore, for any y 2 N0, there is a compact neighborhood B N0 and a holo-morphic section s W B ! f 1.B/, that is, f s D id, such that s.y/ ! .y/ underthe GromovHausdorff convergence of .M; Q!tk / to .X;dX /.

ProofLet A be a countable dense subset of N0, and let K N0 be a compact subset withthe interior intK nonempty. Let Bi be a finite covering of K with small Euclideanballs such that each of the concentric balls B 0i of half radius still cover K . Let si WBi ! f 1.Bi / be sections on Bi , that is, holomorphic maps with f si D id.

Now, we define a map from A \ K D a1; a2; : : : to X . Suppose that thepoint a1 lies inside the ball B 0i , and consider the points si .a1/ inside M . Underthe GromovHausdorff convergence of .M; Q!tk / to .X;dX /, a subsequence of thesepoints converges to a point b1 in X , because the diameter of .M; Q!tk / is uniformlybounded. If a1 also lies inside another ball B 0j , then (1.3) (or also [36, (2.10)]) showsthat d Q!tk .si .a1/; sj .a1// ! 0 when tk ! 0. Thus, by passing to subsequences, bothsi .a1/ and sj .a1/ converge to the same point b1 2 X under the GromovHausdorffconvergence of .M; Q!tk / to .X;dX /. We then define .a1/ D b1. For a2, by repeat-ing the above procedure, we obtain that a subsequence sij .aj /, j D 1; 2, convergesto bj 2 X , j D 1; 2, respectively. Define .a2/ D b2. By repeating this procedureand with a diagonal argument, we can find a subsequence of .M; Q!tk /, denoted by.M; Q!tk / also, such that sij .aj / converges to bj 2 X along the GromovHausdorffconvergence. For any aj 2 A\K , define .aj / D bj .

Now, we prove that W A \ intK ! X is injective. If it is not true, there arey1, y2 2 A \ intK such that y1 y2, and .y1/ D .y2/, which implies thatd Q!tk.si1.y1/; si2.y2// ! 0. If k is a minimal geodesic in .M; Q!tk / connecting si1.y1/and si2.y2/, then

C1length!Nf .k/\K

length Q!tk k \ f 1.K/ d Q!tk si1.y1/; si2.y2/;by (4.1) for a constant C > 0 independent of k. Thus, if f .k/ K for tk 1,

d!N .y1; y2/ C length!Nf .k/

! 0;or, if f .k/\NnK are not empty by passing to a subsequence,

d!N .y1; @K/C d!N .@K;y2/ C length!Nf .k/\K

! 0:In both cases, we obtain contradictions. Thus W A\ intK ! X is injective.

Note that there is an r > 0 such that, for any y 2 intK , the metric ball B!.y; r/is a geodesically convex set, that is, for any y1 and y2 2 B!.y; r/, there is a minimalgeodesic B!.y; r/ connecting y1 and y2, which implies that


d!.y1; y2/ D length!./ 2r:

We take r 1 such that there is a B 0i with B!.y; 2r/ B 0i . If y1; y2 2 A, by Propo-sition 4.6,

dX.y1/; .y2/

D limtk!0

d Q!tksi .y1/; si .y2/

limtk!0

length Q!tksi ./

D length!./ D d!.y1; y2/:

If k is a minimal geodesic in .M; Q!tk / connecting si .y1/ and si .y2/, then (4.20)implies that

e.".tk/=2/length!f .k/\B!.y; 2r/

length Q!tk .k/ ! dX.y1/; .y2/for some function ".t/ ! 0 as t ! 0. If f .k/ B!.y; 2r/ for tk 1 by passing toa subsequence, then

length!f .k/

length!./;since is a minimal geodesic in .N0;!/. If f .k/ \ N0nB!.y; 2r/ is not emptyfor tk 1, then there is a Ny 2 f .k/ \ N0nB!.y; 2r/. Since y1, y2 2 B!.y; r/ andf .k/ connects y1 and y2,

length!f .k/\B!.y; 2r/

d!.y1; Ny/C d!.y2; Ny/ 2r length!./:In both cases,

d!.y1; y2/ D length!./ limtk!0

length!f .k/\B!.y; 2r/

dX

.y1/; .y2/

:

Thus d!.y1; y2/ D dX ..y1/; .y2//; which shows that W .A \ intK;d!/ !.X;dX / is a local isometric embedding. If y1;j and y2;j are two sequences in A\intK such that limj!1 d!.yi;j ; y/ D 0 for i D 1; 2, then limj!1 d!.y1;j ; y2;j / D 0and y1;j ; y2;j B!.y; r/ for j 1. Hence d!.y1;j ; y2;j / D dX ..y1;j /; .y2;j //and d!.yi;j ; yi;jC`/ D dX ..yi;j /; .yi;jC`// for j 1 and any ` 0, whichimplies that .y1;j / and .y2;j / are two Cauchy sequences, and converge toa unique point x 2 X . By defining .y/ D x, extends to a unique map, denoted stillby , from intK to X which is also a local isometric embedding.

Now we prove that .intK/ is an open subset of X . Let x 2 .intK/, i.e. there isa y 2 intK such that .y/ D x, and let x0 2 X with dX .x; x0/ < for a constant 0. Using (5.1), we have

.X/ D V 0.x;D/ D limtk!0

V k.pk;D/ D ZM

!nM :

Proof of Theorem 1.2We prove that X0 X is dense. If this is not true, there is a metric ball BdX .x0; / XnX0. Note that

diamdX .X/ D limtk!0

diam Q!tk .M/ D:

Because of (5.2), we have

BdX .x

0; / .;D/.X/ D $ > 0:

For any compact subset K X0,

.K/ .X/$ D ZM

!nM $


by Lemma 5.2. If BdX .xi ; ri / is a family of metric balls in .X;dX / such that ri < 1, BdX .xi ; 2ri / is a geodesically convex subset of X0, and

Si BdX .xi ; ri / K ,

then Xi

V 0.xi ; ri / DX

i

Zf 1.1.BdX .xi ;ri ///

!nM Zf 1.1.K//

!nM

by Lemma 5.2. Thus

Zf 1.1.K//

!nM lim!0

.K/ D lim!0

infX

i

V 0.xi ; ri /ri <

D .K/:

By taking K large enough such that

.K/ Zf 1.N0/

!nM $

2D

ZM

!nM $

2;

we obtain a contradiction.

Remark 5.3In fact, the same proof shows that .XnX0/ D 0.

Acknowledgments. Most of this work was carried out while the second author wasvisiting the Mathematical Science Center of Tsinghua University in Beijing, whichhe would like to thank for the hospitality extended to him. He is also grateful toD. H. Phong and S.-T. Yau for their support and encouragement and to J. Song formany useful discussions. Some parts of this paper were researched while the thirdauthor was visiting the University of California, San Diego, and the Institut des Hautestudes Scientifiques. He would like to thank both institutions for their hospitality, andhe is also grateful to Professor Xiaochun Rong for helpful discussions.

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GrossDepartment of Mathematics, University of California, San Diego, La Jolla, California 92093,USA; [email protected]

TosattiDepartment of Mathematics, Northwestern University, Evanston, Illinois 60208, USA;[email protected]

ZhangDepartment of Mathematics, Capital Normal University, Beijing 100048, Peoples Republic ofChina; current: Department of Mathematics, University of California, San Diego, La Jolla, Cali-fornia 92093, USA; [email protected]

IntroductionHyperkhler mirror symmetrySemiflat metricsEstimates and smooth convergenceGromovHausdorff convergenceReferencesAuthor's addresses

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