Modular vector bundles on hyperkähler manifolds of Debarre-Voisin type
Alessandro Frassineti, Federico Tufo
TL;DR
This work establishes that on a very general Debarre–Voisin hyperkähler fourfold X ⊂ Gr(6,10), every Schur functor of the restricted quotient bundle Σ_λ Q is modular and polystable, and it provides explicit Ext-group dimensions with concrete 20- and 40-dimensional first-Ext examples. The authors develop and apply a Koszul-complex–based computational pipeline, leveraging Littlewood–Richardson and Borel–Weil–Bott machinery to compute Chern characters and cohomology, yielding both general and case-specific results. They prove that Σ_λ Q is atomic if and only if λ = (m,0,0,0), connecting modularity to extended Mukai vectors and Verbitsky’s hyper-holomorphic framework, and they discuss deformation links to the Beauville–Donagi model. The work also outlines precise combinatorial criteria predicting Ext^1 dimensions and illuminates the potential smoothness of moduli spaces of such modular sheaves, contributing tools and examples for constructing and understanding moduli on hyperkähler manifolds of Debarre–Voisin type.
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.