Degenerations and Stability of Kähler Structures on Calabi--Yau Manifolds

Kefeng Liu; Yang Shen

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arXiv · 2605.18065

Degenerations and Stability of Kähler Structures on Calabi--Yau Manifolds

Kefeng Liu·Yang Shen

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Abstract

In this paper, we study the degeneration and stability of Kähler structures on Calabi--Yau manifolds, namely compact Kähler manifolds with trivial canonical bundles, from the viewpoint of deformation theory and Hodge theory. Using the global deformation theory of Calabi--Yau manifolds together with estimates relating the Weil--Petersson distance and Beltrami differentials, we prove that certain limits of Calabi--Yau manifolds remain Kähler. As applications, we give a new proof of Siu's theorem on the Kählerness of K3 surfaces. We further prove that deformation limits of hyperkähler manifolds with bounded periods remain Kähler, which gives a complete and stronger solution to the conjecture of Soldatenkov--Verbitsky. Finally, we prove that the moduli spaces of stable sheaves on K3 surfaces are hyperkähler manifolds, which gives a complete solution to the conjecture of Perego.

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BibTeX RIS

Kefeng Liu, Yang Shen. 2026-05-18. Degenerations and Stability of Kähler Structures on Calabi--Yau Manifolds. https://arxiv.org/abs/2605.18065

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