Rigid stable vector bundles on hyperkähler varieties of type $K{3}^{[n]}$

Abstract

We prove existence and unicity of slope stable vector bundles on a general polarized hyperkähler (HK) variety of type $K{3}^{[n]}$ 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 in fact we might have listed almost all slope stable rigid projectively hyperholomorphic vector bundles on polarized HK varieties of type $K{3}^{[n]}$ with $20$ moduli.

24 pages. The results for $n=2$, i.e. in dimension 4, are essentially contained in our paper Modular sheaves on hyperkähler varieties, 2022, Algebraic Geometry vol.9, issue 1. Slightly modified according to suggestions of a referee