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BIMSA 代数几何讨论班
BIMSA 代数几何讨论班
Deformation of Kähler Structures via Beltrami Differentials: Stability and Degenerations
Deformation of Kähler Structures via Beltrami Differentials: Stability and Degenerations
演讲者
刘克峰
时间
2026年05月28日 10:00 至 11:00
地点
A6-101
线上
Zoom 518 868 7656
(BIMSA)
摘要
How do Kähler structures behave under deformations and degenerations of compact complex manifolds? We present a new Hodge-theoretic approach to this question, built on deformation theory and integrable Beltrami differentials.
The central idea is to use explicit sections of Hodge bundles to track the variation of $(p,p)$-classes across a family. This yields both a stability theorem — showing that the Kähler property persists on large regions of the deformation space — and a degeneration theorem controlling Kähler structures at limits.
As applications, we focus on Calabi–Yau manifolds, particularly hyperkähler manifolds. We prove that deformation limits of hyperkähler manifolds with bounded periods remain Kähler. This gives a new, purely deformation-theoretic proof of Siu's theorem that every K3 surface is Kähler, and resolves conjectures of Soldatenkov–Verbitsky and Perego in stronger forms.
The central idea is to use explicit sections of Hodge bundles to track the variation of $(p,p)$-classes across a family. This yields both a stability theorem — showing that the Kähler property persists on large regions of the deformation space — and a degeneration theorem controlling Kähler structures at limits.
As applications, we focus on Calabi–Yau manifolds, particularly hyperkähler manifolds. We prove that deformation limits of hyperkähler manifolds with bounded periods remain Kähler. This gives a new, purely deformation-theoretic proof of Siu's theorem that every K3 surface is Kähler, and resolves conjectures of Soldatenkov–Verbitsky and Perego in stronger forms.