Rank 4 stable vector bundles on hyperk\"ahler fourfolds of Kummer type

TL;DR

This paper extends results on stable vector bundles to hyperk"ahler fourfolds of Kummer type, constructing a unique, rigid, slope stable rank 4 bundle with specified Chern classes under certain conditions.

Contribution

It constructs and characterizes a unique stable, rigid rank 4 vector bundle on general polarized hyperk"ahler fourfolds of Kummer type with specific numerical conditions.

Findings

Existence of a unique slope stable vector bundle with given invariants.

The bundle is rigid, implying no non-trivial deformations.

Conditions on polarization ensure the bundle's existence and uniqueness.

Abstract

We partially extend to hyperk\"ahler fourfolds of Kummer type the results that we have proved regarding stable rigid vector bundles on hyperk\"ahler (HK) varieties of type $K3^{[n]}$. Let $(M,h)$ be a general polarized HK fourfold of Kummer type such that $q_M(h)\equiv -6\pmod{16}$ and the divisibility of $h$ is $2$, or $q_M(h)\equiv -6\pmod{144}$ and the divisibility of $h$ is $6$. We show that there exists a unique (up to isomorphism) slope stable vector bundle $\cal F$ on $M$ such that $r({\cal F})=4$, $ c_1({\cal F})=h$, $\Delta({\cal F})=c_2(M)$. Moreover $\cal F$ is rigid. One of our motivations is the desire to describe explicitly a locally complete family of polarized HK fourfolds of Kummer type.