Modular sheaves on hyperkähler varieties

A torsion-free sheaf on a hyperkaehler manifold X is modular if its discriminant satisfies a certain condition; for example, this is the case if it is a multiple of the second Chern class of X. The definition is tailor-made for torsion-free sheaves on a polarized hyperkaehler variety (X,h) which deform to all small deformations of (X,h). For hyperkaehler varieties which are deformations of the Hilbert square of a K3 surface, we prove an existence and uniqueness result for slope-stable vector bundles with certain ranks, and first two Chern classes. As a consequence, we get the uniqueness up to isomorphism of the tautological quotient rank 4 vector bundle on the variety of lines on a generic cubic 4- dimensional hypersurface, and on the Debarre–Voisin variety associated with a generic (complex) skew-symmetric trilinear form in 10 variables. The last result implies that the period map from the moduli space of Debarre–Voisin varieties to the relevant period space is birational.