Compactification of metric moduli space of $K3$ surfaces

TL;DR

This paper proves a conjecture linking algebraic and metric compactifications of K3 surfaces, providing a classification of their Gromov--Hausdorff limits under fixed complex structure or polarization.

Contribution

It establishes an algebraic description of the Gromov--Hausdorff compactification for hyperk"ahler K3 surfaces, confirming a conjecture and classifying their metric limits.

Findings

Confirmed Odaka--Oshima conjecture

Provided algebraic description of metric compactification

Classified Gromov--Hausdorff limits of hyperk"ahler K3 surfaces

Abstract

We prove a conjecture of Odaka--Oshima, which says that there is an algebraic description of the Gromov--Hausdorff compactification of all unit-diameter hyperk\"ahler metrics on K3 surfaces. As a corollary, we obtain a classification of the Gromov--Hausdorff limits of those hyperk\"ahler K3 surfaces with a fixed complex structure or with a fixed polarization.