Compactification of metric moduli space of $K3$ surfaces

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.

💡 Research Summary

The paper establishes a precise algebraic description of the Gromov‑Hausdorff (GH) compactification of the space of unit‑diameter hyperkähler metrics on K3 surfaces, confirming a conjecture of Odaka–Oshima. The authors consider the metric moduli space
(M={ \text{isometry classes of unit‑diameter hyperkähler metrics on K3 surfaces}})
endowed with the GH topology. By Gromov’s pre‑compactness theorem, its closure (\overline{M}) is compact. Earlier works (