Degenerations and Stability of Kähler Structures on Calabi--Yau Manifolds
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909054034509824 |
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| author | Liu, Kefeng Shen, Yang |
| author_facet | Liu, Kefeng Shen, Yang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18065 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Degenerations and Stability of Kähler Structures on Calabi--Yau Manifolds Liu, Kefeng Shen, Yang Algebraic Geometry 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. |
| title | Degenerations and Stability of Kähler Structures on Calabi--Yau Manifolds |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2605.18065 |