Collapsing Ricci-flat metrics on elliptic K3 surfaces

TL;DR

This paper constructs a family of Ricci-flat Kähler metrics on elliptic K3 surfaces that collapse to a base sphere with a specific metric, providing detailed descriptions near singular fibers without restrictions.

Contribution

It introduces a new construction of collapsing Ricci-flat metrics on elliptic K3 surfaces with detailed local descriptions near all types of singular fibers.

Findings

Metrics converge to the McLean metric on the base

Curvature remains bounded away from singular fibers

Provides explicit local descriptions near singular fibers

Abstract

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.