The Kummer Construction of Calabi-Yau and Hyper-Kähler Metrics on the $K3$ Surface, and Large Families of Volume Non-collapsed Limiting Compact Hyper-Kähler Orbifolds

Abstract: Right after Yau's resolution of the Calabi conjecture in the late 1970s, physicists Page and Gibbons-Pope conjectured that one may approximate Ricci-flat K\"{a}hler metrics on the $K3$ surface with metrics having "almost special holonomy" constructed via "resolving" the $16$ orbifold singularities of a flat $\mathbb{T}^{4}/\mathbb{Z}_{2}$ with Eguchi-Hanson metrics. Constructions of such metrics with special holonomy from such a "gluing" construction of approximate special holonomy metrics have since been called "Kummer constructions" of special holonomy metrics, and their proposal was rigorously carried out in the 1990s by Kobayashi and LeBrun-Singer, and in the 2010s by Donaldson. In this paper, we provide two new rigorous proofs of Page-Gibbons-Pope's proposal based on singular perturbation and weighted function space analysis. Each proof is done from a different perspective: * solving the complex Monge-Ampere equation (Calabi-Yau) * perturbing closed definite triples (hyper-K\"{a}hler) Both proofs yield Eguchi-Hanson metrics as ALE bubbles/rescaled limits. Moreover, our analysis in the former perspective yields estimates which improve Kobayashi's estimates, and our analysis in the latter perspective results in the construction of large families of Ricci-flat K\"{a}hler metrics on the $K3$ surface, yielding the full $58$ dimensional moduli space of such. Finally, as a byproduct of our analysis, we produce a plethora of large families of compact hyper-K\"{a}hler orbifolds which all arise as volume non-collapsed Gromov-Hausdorff limit spaces of the aforementioned constructed large families of Ricci-flat K\"{a}hler metrics on the $K3$ surface. Moreover, these compact hyper-K\"{a}hler orbifolds are explicitly exhibited as points in the "holes" of the moduli space of Ricci-flat K\"{a}hler-Einstein metrics on $K3$ under the period map.