A characterization of compact locally conformally hyperk\"ahler manifolds

TL;DR

This paper provides an equivalent characterization of compact locally conformally hyperk"ahler manifolds using a special complex two-form, extending Beauville's theorem to a conformal setting.

Contribution

It introduces a new equivalent definition for these manifolds based on a nondegenerate complex two-form with specific properties.

Findings

Characterization of compact locally conformally hyperk"ahler manifolds

Extension of Beauville's theorem to conformal case

Identification of a natural complex two-form as a key property

Abstract

We give an equivalent definition of compact locally conformally hyperk\"ahler manifolds in terms of the existence of a nondegenerate complex two-form with natural properties. This is a conformal analogue of Beauville's theorem stating that a compact K\"ahler manifold admitting a holomorphic symplectic form is hyperk\"ahler.