Collapsing Ricci-flat metrics on elliptic K3 surfaces
Collapsing Ricci-flat metrics on elliptic K3 surfaces
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DOI：
10.4310/cag.2020.v28.n8.a9
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发表时间：
2019-10
影响因子：
0.7
通讯作者：
Gao Chen;Jeff A. Viaclovsky;Ruobing Zhang
中科院分区：
文献类型：
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作者：
Gao Chen;Jeff A. Viaclovsky;Ruobing Zhang
对于任意椭圆K3曲面$\mathfrak{F}：\mathcal{K} \rightarrow \mathbb{P}^1$，我们构造了一族坍缩的Ricci平坦K\“ahler度量，使得曲率一致有界远离奇异纤维，并且Gromov-Hausdorff将其限制为$\mathbb{P}^1$，并配备了姆克林度量.这种类型的坍缩有一些众所周知的例子，但我们构造的关键点是，我们可以另外给出每种类型奇异纤维附近的度量退化的精确描述，而对奇异纤维的类型没有任何限制。
For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat K\"ahler 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.