arXiv:1304.1820v1 [math.DG] 5 Apr 2013GROMOV-HAUSDORFF COLLAPSING OF CALABI-YAUMANIFOLDSMARK GROSS∗, VALENTINO TOSATTI†, AND YUGUANG ZHANG‡Abstract.This paper is a sequel to . We further study Gromov-Hausdorff collapsing limits of Ricci-flat K¨ahler metrics on abelian fiberedCalabi-Yau manifolds. Firstly, we show that in the same setup as ,if the dimension of the base manifold is one, the limit metric space ishomeomorphic to the base manifold. Secondly, if the fibered Calabi-Yaumanifolds are Lagrangian fibrations of holomorphic symplectic mani-folds, the metrics on the regular parts of the limits are special K¨ahlermetrics. By combining these two results, we extend  to any fiberedprojective K3 surface without any assumption on the type of singularfibers.1.IntroductionIn this paper we continue our study in  of the structure of collapsedGromov-Hausdorff limits of Ricci-flat K¨ahler metrics on compact Calabi-Yau manifolds.LetMbe a projective Calabi-Yau manifold of complexdimensionm, with Ω a nowhere vanishing holomorphicm-form onM. LetNbe a projective manifold of dimension 0<n<m, andf:M→Nbea holomorphic fibration (i.e., a surjective holomorphic map with connectedfibers) whose general fibre is an abelian variety. Letαbe an ample class onM, and letN0⊂Nbe the Zariski open subset such that, for anyy∈N0,My=f−1(y) is smooth (and therefore Calabi-Yau).LetD=N\N0bethe discriminant locus of the mapf. Letα0be an ample class onN, and˜ωt∈f∗α0+tαbe the Ricci-flat K¨ahler metric given by Yau’s Theorem fort∈(0,1], which satisfies the complex Monge-Amp`ere equation(1.1)˜ωmt=cttm−n(−1)m22Ω∧Ω.By  and , ˜ωtconverges smoothly tof∗ωonf−1(K) for any compactK⊂N0ast→0, and onN0, whereωis the K¨ahler metric onN0withRic(ω) =ωW Pobtained in  and  (see also ), andωW Pis a Weil-Petersson semi-positive form onN0coming from the variation of the complex structures ofthe fibersMy. Furthermore, the Ricci-flat metrics ˜ωthave locally uniformly∗Supported in part by NSF grant DMS-1105871.†Supported in part by a Sloan Research Fellowship and NSF grant DMS-1236969.‡Supported in part by NSFC-11271015.1
2M. GROSS, V. TOSATTI, AND Y. ZHANGbounded curvature onf−1(N0).Thanks to  and , the diameter ofthe metrics ˜ωtsatisfiesdiam˜ωt(M)lessorequalslantC,for a constantC>0 independent oft. Gromov’s pre-compactness theorem(cf. ) then implies that for any sequencetk→0, a subsequence of (M,˜ωtk)converges to a compact metric space (X,dX) in the Gromov-Hausdorff sense(a priori different subsequences could result in non-isometric limits). In ourearlier work  we proved that (N0,ω) can be locally isometrically embed-ded into (X,dX), with open dense imageX0⊂X, via a homeomorphismφ:N0→X0. The following questions remain open (see [36, 37]).
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Y. Zhang, Symplectic manifold, M. GROSS, calabi-yau manifolds