Modular sheaves on hyperk\"ahler varieties

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.