1304.1820v1 - arXiv:1304.1820v1[math.DG 5 Apr 2013 GROMOV-HAUSDORFF COLLAPSING OF CALABI-YAU MANIFOLDS MARK GROSS VALENTINO TOSATTI AND YUGUANG ZHANG

1304.1820v1 - arXiv:1304.1820v1[math.DG 5 Apr 2013...

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arXiv:1304.1820v1 [math.DG] 5 Apr 2013 GROMOV-HAUSDORFF COLLAPSING OF CALABI-YAU MANIFOLDS MARK GROSS , VALENTINO TOSATTI , AND YUGUANG ZHANG Abstract. This paper is a sequel to [12]. We further study Gromov- Hausdorff collapsing limits of Ricci-flat K¨ ahler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as [12], if the dimension of the base manifold is one, the limit metric space is homeomorphic to the base manifold. Secondly, if the fibered Calabi-Yau manifolds are Lagrangian fibrations of holomorphic symplectic mani- folds, the metrics on the regular parts of the limits are special K¨ ahler metrics. By combining these two results, we extend [13] to any fibered projective K3 surface without any assumption on the type of singular fibers. 1. Introduction In this paper we continue our study in [12] of the structure of collapsed Gromov-Hausdorff limits of Ricci-flat K¨ ahler metrics on compact Calabi- Yau manifolds. Let M be a projective Calabi-Yau manifold of complex dimension m , with Ω a nowhere vanishing holomorphic m -form on M . Let N be a projective manifold of dimension 0 <n<m , and f : M N be a holomorphic fibration (i.e., a surjective holomorphic map with connected fibers) whose general fibre is an abelian variety. Let α be an ample class on M , and let N 0 N be the Zariski open subset such that, for any y N 0 , M y = f 1 ( y ) is smooth (and therefore Calabi-Yau). Let D = N \ N 0 be the discriminant locus of the map f . Let α 0 be an ample class on N , and ˜ ω t f α 0 + be the Ricci-flat K¨ ahler metric given by Yau’s Theorem [40] for t (0 , 1], which satisfies the complex Monge-Amp` ere equation (1.1) ˜ ω m t = c t t m n ( 1) m 2 2 Ω Ω . By [35] and [12], ˜ ω t converges smoothly to f ω on f 1 ( K ) for any compact K N 0 as t 0, and on N 0 , where ω is the K¨ ahler metric on N 0 with Ric( ω ) = ω W P obtained in [35] and [29] (see also [28]), and ω W P is a Weil-Petersson semi- positive form on N 0 coming from the variation of the complex structures of the fibers M y . Furthermore, the Ricci-flat metrics ˜ ω t have 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
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2 M. GROSS, V. TOSATTI, AND Y. ZHANG bounded curvature on f 1 ( N 0 ). Thanks to [34] and [41], the diameter of the metrics ˜ ω t satisfies diam ˜ ω t ( M ) lessorequalslant C, for a constant C> 0 independent of t . Gromov’s pre-compactness theorem (cf. [10]) then implies that for any sequence t k 0, a subsequence of ( M, ˜ ω t k ) converges to a compact metric space ( X,d X ) in the Gromov-Hausdorff sense (a priori different subsequences could result in non-isometric limits). In our earlier work [12] we proved that ( N 0 ) can be locally isometrically embed- ded into ( X,d X ), with open dense image X 0 X , via a homeomorphism φ : N 0 X 0 . The following questions remain open (see [36, 37]).
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