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

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.