Modular vector bundles with and without moduli

TL;DR

This paper studies modular vector bundles on specific algebraic varieties, showing their stability properties, rigidity conditions, and computing Ext-groups, revealing new stable bundles with large Ext groups on K3^[2] manifolds.

Contribution

It proves that certain Schur functor bundles on Fano varieties of lines are modular and slope polystable, and characterizes when they are rigid and stable, also computing their Ext-groups.

Findings

Schur functor bundles are modular and slope polystable on the Fano variety.

Rigid bundles are exactly those that are atomic and slope stable.

Existence of new stable modular bundles on K3^[2] manifolds with 40-dimensional Ext^1 groups.

Abstract

If $X\subset\operatorname{Gr}(2,6)$ is the Fano variety of lines of a smooth cubic fourfold, then we show that the restriction to $X$ of any Schur functor of the tautological quotient bundle is modular and slope polystable. Moreover it is atomic if and only if it is rigid, in which case it is also slope stable. We further compute the Ext-groups of such bundles in infinitely many cases, showing in particular the existence of new modular vector bundles on manifolds of type $\operatorname{K3}^{[2]}$ that are slope stable and whose $\operatorname{Ext}^1$-group is 40-dimensional.