Rigid stable vector bundles on hyperk\"ahler varieties of type $K3^{[n]}$

TL;DR

This paper establishes the existence and uniqueness of certain slope stable vector bundles on general polarized hyperk"ahler varieties of type $K3^{[n]}$, focusing on specific invariants and conditions.

Contribution

It proves the existence and unicity of slope stable vector bundles with particular invariants on generic polarized hyperk"ahler varieties of type $K3^{[n]}$, advancing understanding of their moduli.

Findings

Existence of slope stable vector bundles under specific conditions

Uniqueness of these vector bundles on general polarized HK varieties

Almost all such rigid projectively hyperholomorphic bundles are characterized

Abstract

We prove existence and unicity of slope stable vector bundles on a general polarized hyperk\"ahler (HK) variety of type $K3^{[n]}$ 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 $K3^{[n]}$ with $20$ moduli.