New constructions of (twisted) hyperholomorphic vector bundles on hyperkähler manifolds and their applications

Hyperholomorphic vector bundles on hyperkähler manifolds are special vector bundles with remarkable deformation properties. These objects have found applications in the study of algebraic cycles and derived categories for hyperkähler manifolds of K3^{[n]} type, i.e. those which are deformation equivalent to Hilbert schemes of points on K3 surfaces. I will present a new construction of hyperholomorphic bundles on hyperkähler manifolds of K3^{[n]}-type and OG10-type in the presence of a Lagrangian fibration, which allows many of these applications to be recast in the setting of O’Grady’s hyperkähler tenfolds. This is joint work with Moritz Hartlieb.