Comparison with Markman’s hyperholomorphic bundles

Determine whether the twisted hyperholomorphic vector bundle Q constructed from transverse Lagrangian fibrations is deformation equivalent along diagonal twistor lines to one of Markman’s vector bundles U^[n].

Background

The paper constructs a twisted hyperholomorphic vector bundle Q on a product of hyperkähler manifolds of K- or OG-type by convolving twisted Poincaré sheaves associated with two transverse Lagrangian fibrations.

Markman previously constructed hyperholomorphic bundles U^[n] for products of hyperkähler manifolds of K-type. The authors explicitly leave unresolved whether their new construction yields bundles deformation equivalent to Markman’s bundles along diagonal twistor lines.