Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces

TL;DR

This paper constructs families of Ricci-flat Kähler metrics on K3 surfaces that collapse to an interval, revealing bubble limits like Tian-Yau and Taub-NUT metrics, and describes the fiber structures involved.

Contribution

It introduces new collapsing Ricci-flat metrics on K3 surfaces with detailed fiber and bubble analysis, expanding understanding of metric degenerations.

Findings

Collapse to an interval with specific fiber types

Occurrence of Tian-Yau and Taub-NUT metrics as bubbles

Existence of a continuous surjective map from K3 to the interval

Abstract

We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.