We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general hypotheses. Assuming a result in pluripotential theory, we prove a uniform fibre diameter bound for a Lefschetz K3 fibred Calabi-Yau 3-fold, which reduces the study of the collapsing metric to a locally non-collapsed situation, and we identify the Gromov-Hausdorff limit of the rescaled neighbourhood of the singular fibre.

中文翻译：

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关于崩溃的卡拉比-丘纤维

我们开发了一些技术来研究 Calabi-Yau 度量对纤维化总空间的绝热限制行为，并通过对奇点类型施加限制来获得对奇异纤维附近的强控制。在相当普遍的假设下，我们证明了直到奇异纤维的度量的统一下限。假设多能理论的结果，我们证明了 Lefschetz K3 纤维 Calabi-Yau 3 倍的均匀纤维直径界限，这将塌陷度量的研究减少到局部非塌陷情况，并且我们确定了 Gromov-Hausdorff 极限的奇异纤维的重新缩放的邻域。