Modular vector bundles on hyperk\"ahler manifolds of Debarre-Voisin type

TL;DR

This paper studies the properties of certain vector bundles on a special class of hyperk"ahler manifolds, showing their modularity, stability, and computing their Ext-groups, revealing new structural insights.

Contribution

It proves that all Schur functors of a specific quotient bundle are modular and polystable, and characterizes atomic bundles as symmetric powers, with explicit Ext-group computations.

Findings

All Schur functors of the quotient bundle are modular and polystable.

Atomic bundles correspond to symmetric powers of the quotient bundle.

Examples with 20 and 40 dimensional Ext-groups are constructed.

Abstract

Let X be a very general Debarre-Voisin fourfold. In this article, we prove that all the Schur functors of the restriction of the quotient bundle of Gr(6,10) to X are modular and polystable vector bundles. We also show that such bundles are atomic if and only if correspond to the symmetric power of the restriction of the quotient bundle. Moreover, we compute the Ext-groups of different modular vector bundles on X, and we find examples with 20 and 40 dimensional first Ext-group.