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

TL;DR

The paper provides two rigorous Kummer constructions of Ricci-flat metrics on the K3 surface, one via solving the complex Monge–Ampère equation (Calabi–Yau perspective) and another via perturbing closed definite triples (hyper-Kähler perspective). Both approaches glue in Eguchi–Hanson ALE bubbles at the 16 singular points of ${f T}^4/{f Z}_2$, and use singular perturbation together with weighted Hölder spaces to obtain uniform invertibility of the linearized operators, yielding genuine Ricci-flat metrics for small gluing parameters. The Calabi–Yau analysis produces a 1-parameter family of Ricci-flat Kähler metrics in a fixed complex structure with explicit Gromov–Hausdorff convergence to the flat orbifold and ALE bubbles; the hyper-Kähler analysis yields a 58-parameter moduli family of HK metrics with a similar bubble structure, and a rich picture of volume-noncollapsed HK orbifolds. In addition, the work identifies a large class of compact HK orbifolds arising as limits, situates them inside codimension-3 holes of the period map, and demonstrates iterated volume-noncollapsed degenerations, thereby evidencing sharpness of the codimension-4 and codimension-3 phenomena in the period map boundary.

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 Kähler 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-Kähler) 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 Kähler 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-Kähler orbifolds which all arise as volume non-collapsed Gromov-Hausdorff limit spaces of the aforementioned constructed large families of Ricci-flat Kähler metrics on the $K3$ surface. Moreover, these compact hyper-Kähler orbifolds are explicitly exhibited as points in the "holes" of the moduli space of Ricci-flat Kähler-Einstein metrics on $K3$ under the period map.

Theorems & Definitions (88)

• theorem 1.1: Main Theorem I, see Theorem \ref{['maintheorem1']}
• theorem 1.2: Main Theorem II, see Theorem \ref{['maintheorem2']}
• remark 1.3
• remark 1.4
• remark 1.5
• Definition 3.1
• remark 3.2
• remark 3.3
• remark 3.4
• Proposition 3.5