Modular sheaves on hyperk\"ahler varieties

TL;DR

This paper introduces the concept of modular sheaves on hyperk"ahler varieties, proves existence and uniqueness results for certain stable vector bundles, and applies these to specific geometric cases, establishing a birational period map.

Contribution

It defines modular sheaves on hyperk"ahler varieties and proves existence and uniqueness of stable modular vector bundles in specific deformation classes.

Findings

Uniqueness of stable modular vector bundles on hyperk"ahler varieties of type K3^{[2]}.

Application to the uniqueness of tautological quotient bundles on certain hypersurfaces.

Establishment of a birational period map for Debarre-Voisin varieties.

Abstract

A torsion free sheaf on a hyperk\"ahler variety $X$ is modular if the discriminant satisfies a certain condition, for example if it is a multiple of $c_2(X)$ the sheaf is modular. The definition is taylor made for torsion-free sheaves on a polarized hyperk\"ahler variety (X,h) which deform to all small deformations of (X,h). For hyperk\"ahlers deformation equivalent to $K3^{[2]}$ we prove an existence and uniqueness result for slope-stable modular vector bundles with certain ranks, $c_1$ and $c_2$. As a consequence we get uniqueness up to isomorphism of the tautological quotient rank $4$ vector bundles on the variety of lines on a generic cubic $4$-dimensional hypersurface, and on the Debarre-Voisin variety associated to a generic skew-symmetric $3$-form on a $10$-dimensional complex vector space. The last result implies that the period map from the moduli space of Debarre-Voisin varieties to the relevant period space is birational.