2014
DOI: 10.4310/mrl.2014.v21.n4.a15
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Triviality of fibered Calabi-Yau manifolds without singular fibers

Abstract: In this note we show that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a holomorphic fiber bundle. In the algebraic case, the fibration becomes trivial after a finite base change.Comment: 11 pages; final versio

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Cited by 19 publications
(20 citation statements)
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“…However it seems hopeless to pursue this strategy in higher dimensions when there are singular fibers, since in general there is no good local model metric near the singularities. Note that such fibrations with S = ∅ are very special, thanks to the results in [59].…”
Section: 2mentioning
confidence: 99%
“…However it seems hopeless to pursue this strategy in higher dimensions when there are singular fibers, since in general there is no good local model metric near the singularities. Note that such fibrations with S = ∅ are very special, thanks to the results in [59].…”
Section: 2mentioning
confidence: 99%
“…An interesting application revolves around Calabi-Yau manifolds. Specifically, we apply Proposition 4.1 to the following result of Tosatti-Zhang: Theorem 5.2 (Theorems 1.2 and 1.3 in [TZ14]). Suppose π : X → Y is a holomorphic submersion with connected fiber F satisfying one of the following:…”
Section: Polar Foliations Toric Symmetry and Calabi-yau Bundlesmentioning
confidence: 99%
“…To the authors' knowledge, Theorem H is the first result on prescribing scalar curvature functions on non-compact fibre bundles over Calabi-Yau manifolds. On the other hand, Tosatti-Zhang [TZ14] proved that homolorphic submersions from compact Calabi-Yau manifolds are trivial, up to covering. We combine Theorem H to this result of Tosatti-Zhang to obtain: Corollary J.…”
Section: Introductionmentioning
confidence: 99%
“…The phenomenon which appears in (iii) also occurs for higher-dimensional fiber spaces. Indeed, it was shown in [69,71] that if f : X → Y is a holomorphic submersion with connected fibers from a Calabi-Yau manifold X to a Kähler manifold Y , then f is a fiber bundle.…”
Section: Let Us Give Some Examplesmentioning
confidence: 99%
“…The partial second-order estimate is motivated by the study of collapsing problems in Kähler geometry (see, e.g., [1,6,14,22,23,28,29,32,33,34,35,41,42,47,59,53,54,55,56,58,63,64,66,67,68,69,71,74,80]), as well as the study of canonical metrics in Kähler geometry and the behavior of the Kähler-Ricci flow. More specifically, the estimate in Theorem A plays a crucial role in establishing the following conjectural picture for collapsed Gromov-Hausdorff limits of Ricci-flat Kähler metrics which were originally proposed in [66,67], inspired by [24,38,39] and has been intensively studied since:…”
Section: Introductionmentioning
confidence: 99%