Limits of Hyperkähler Manifolds and Moduli Spaces of Sheaves on K3 Surfaces

Abstract

We prove that a compact strongly Gauduchon manifold which is a limit of compact hyperkähler manifolds is bimeromorphic to a compact hyperkähler manifold. This settles a case of a conjecture of Soldatenkov and Verbitsky. A closed completion of a Gauduchon metric bounds the Dirichlet energies of nearby biholomorphisms. This bound yields a meromorphic limit map, which has degree one and is therefore a bimeromorphism onto a compact hyperkähler manifold. We apply this result to moduli spaces of stable torsion-free coherent sheaves on K3 surfaces. A Bott--Chern diagonal formula gives the $\partial \bar{\partial }$-lemma, and positivity of determinant classes gives Kählerness, proving Perego's conjecture.

arXiv commentSubstantially revised; title changed. The result on moduli spaces of stable sheaves on K3 surfaces is retained, with a new proof. The general Kahlerness assertions in the previous version are no longer claimed. The present version establishes a bimeromorphic hyperkahler model for limits satisfying strongly Gauduchon