Degenerations and Stability of K\"ahler Structures on Calabi--Yau Manifolds

TL;DR

This paper investigates how K"ahler structures on Calabi--Yau manifolds behave under degeneration, establishing conditions for stability and applying results to K3 and hyperk"ahler manifolds.

Contribution

It provides new proofs and stronger results on the stability of K"ahler structures during degeneration, including solutions to several conjectures.

Findings

Limits of Calabi--Yau manifolds remain K"ahler under certain conditions.

Deformation limits of hyperk"ahler manifolds with bounded periods are K"ahler.

Moduli spaces of stable sheaves on K3 surfaces are hyperk"ahler.

Abstract

In this paper, we study the degeneration and stability of K\"ahler structures on Calabi--Yau manifolds, namely compact K\"ahler 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 K\"ahler. As applications, we give a new proof of Siu's theorem on the K\"ahlerness of K3 surfaces. We further prove that deformation limits of hyperk\"ahler manifolds with bounded periods remain K\"ahler, 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 hyperk\"ahler manifolds, which gives a complete solution to the conjecture of Perego.