Collapsing K3 surfaces, Tropical geometry and Moduli compactifications of Satake, Morgan-Shalen type

TL;DR

This paper develops a moduli-theoretic approach to understand the collapsing behavior of Ricci-flat Kähler metrics, particularly for K3 surfaces, using compactifications inspired by Morgan-Shalen and Satake, and studies their Gromov-Hausdorff limits.

Contribution

It introduces a new framework connecting moduli space compactifications with metric collapsing phenomena for Ricci-flat Kähler metrics.

Findings

Describes the collapsing of K3 surfaces to lower-dimensional spaces.

Provides a moduli-theoretic interpretation of metric limits.

Analyzes Gromov-Hausdorff limits of hyperKähler metrics.

Abstract

We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study the Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters, especially for K3 surfaces.