| Résume | Résumé : Since moduli of sheaves on K3 surfaces play a key role in Algebraic
Geometry, and since K3's are the two dimensional
hyperkähler (HK) manifolds, it is natural to investigate moduli of sheaves on
higher dimensional HK's. We propose to focus
attention on (coherent) torsion free sheaves on a HK variety X whose
discriminant in H^4(X) satisfies a certain condition. These are the modular
sheaves
of the title. For example a sheaf whose discriminant is a multiple of c_2(X)
is modular.
For HK's which are deformations of the Hilbert square of a K3 we prove an
existence and uniqueness result for slope-stable 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 bundle 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-tensor
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. |
