Rigid stable vector bundles on hyperkähler varieties of type $K3^{[n]}$

Abstract

We prove existence and unicity results for slope stable vector bundles on a general polarized hyperkähler (HK) variety of type K3 rns with certain discrete invariants, provided the rank and the first two Chern classes of the vector bundle satisfy certain equalities. The latter hypotheses at first glance appear to be quite restrictive, but we suspect that we have listed almost all slope stable rigid projectively hyperholomorphic vector bundles on polarized HK varieties of type K3 rns with 20 moduli.