The harmonic 2-forms on K3 surfaces converging to a flat 4-dimensional orbifold

2025/12/16 by Hattori, Kota
#53C26 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.14125

Abstract

In this article, we study the asymptotic behavior of harmonic 2-forms on K3 surfaces with Ricci-flat Kähler metrics, where metrics converge to the quotient of a flat 4-torus by a finite group action. We can show that the space of anti-self-dual harmonic 2 forms decomposes into two subspaces: one converges to the flat 2-forms on the quotient of the torus, while the other converges to the first Chern forms of anti-self-dual connections on ALE spaces.

Citations

• Remarks on Kähler Orbifolds of non-negative Ricci curvature
• On asymptotic behavior of the second Chern forms on degenerating Kähler-Einstein surfaces
• 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
• Spectral convergence under bounded Ricci curvature
• Calabi-Yau metrics on Kummer surfaces as a model glueing problem
• Blowing up and desingularizing constant scalar curvature Kähler manifolds
• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator

Related

• Products of harmonic forms and rational curves
• Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces
• Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, and Almost-Kaehler Geometry
• ALE Ricci-flat Kahler metrics and deformations of quotient surface\n singularities
• Harmonic forms on ALE Ricci-flat 4-manifolds