2019/12/05 by O'Grady, Kieran G. · 2 citations
#14J42 #14J60 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1912.02659
A torsion free sheaf on a hyperkähler variety X is modular if the discriminant satisfies a certain condition, for example if it is a multiple of c2(X) the sheaf is modular. The definition is taylor made for torsion-free sheaves on a polarized hyperkähler variety (X,h) which deform to all small deformations of (X,h). For hyperkählers deformation equivalent to K3[2] we prove an existence and uniqueness result for slope-stable modular vector bundles with certain ranks, c1 and c2. 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.