Compactification of metric moduli space of $K3$ 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 hyperkähler metrics on K3 surfaces. As a corollary, we obtain a classification of the Gromov--Hausdorff limits of those hyperkähler K3 surfaces with a fixed complex structure or with a fixed polarization.