For any elliptic K3 surface $\mathfrak{F} : \mathcal{K} \to \mathbb{P}^1$, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov–Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can additionally give a precise description of the metric degeneration near each type of singular fiber, without any restriction on the types of singular fibers.

中文翻译：

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椭圆K3曲面上的折叠Ricci-flat度量

对于任何椭圆形K3曲面$ \ mathfrak {F}：\ mathcal {K} \ to \ mathbb {P} ^ 1 $，我们构造了一系列可折叠的Ricci-flatKähler度量，以使曲率从奇异纤维均匀地定界，以及配备了McLean度量标准的Gromov–Hausdorff限制为$ \ mathbb {P} ^ 1 $。有这种类型的崩溃的众所周知的例子，但是我们构造的关键是我们可以在每种类型的奇异纤维附近另外精确地描述度量退化，而对奇异纤维的类型没有任何限制。