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arxiv: 1910.11321 · v1 · pith:KXYU2EZEnew · submitted 2019-10-24 · 🧮 math.DG · math.AG

Collapsing Ricci-flat metrics on elliptic K3 surfaces

classification 🧮 math.DG math.AG
keywords collapsingsingularellipticfibersmathbbmetricmetricsricci-flat
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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.

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