Description:
<jats:title>Abstract</jats:title><jats:p>Let (<jats:italic>X, H</jats:italic>) be a polarized smooth projective surface satisfying<jats:italic>H</jats:italic><jats:sup>1</jats:sup>(<jats:italic>Χ, O<jats:sub>Χ</jats:sub></jats:italic>) = 0, and let<jats:italic>Ƒ</jats:italic>be either a rank 1 torsion-free sheaf or a rank 2<jats:italic>μ<jats:sub>H</jats:sub></jats:italic>-stable vector bundle on<jats:italic>Χ</jats:italic>. Assume that<jats:italic>c</jats:italic><jats:sub>1</jats:sub>(<jats:italic>Ƒ</jats:italic>) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheaf<jats:italic>Ƒ</jats:italic><jats:sup>[2]</jats:sup>associated with<jats:italic>Ƒ</jats:italic>on the Hilbert square<jats:italic>Χ</jats:italic><jats:sup>[2]</jats:sup>is<jats:italic>μ</jats:italic>-stable with respect to a certain polarization.</jats:p>