Restriction of the Poincaré Bundle to a Calabi-yau Hypersurface

Let X be a compact connected Riemann surface of genus g, where g ≥ 3. Denote by Mξ := M(n, ξ) the moduli space of stable vector bundles over X of rank n and fixed determinant ξ. If the degree deg(ξ) and the rank n are coprime, then there is a universal family of vector bundles, U , over X parametrized by Mξ. This family is unique up to tensoring by a line bundle that comes from Mξ. We fix one universal family over X × Mξ and call it the Poincaré bundle. For any x ∈ X, let Ux denote the vector bundle over Mξ obtained by restricting U to x ×Mξ. It is known that U (see [BBN]) and Ux (see [NR] and [Ty]) are stable vector bundles with respect to any polarization on X ×Mξ and Mξ respectively. A smooth anti-canonical divisor D on Mξ is an example of a Calabi-Yau variety, i.e., it is connected and simply connected with trivial canonical line bundle. The Calabi-Yau varieties are of interest both in string theory and in algebraic geometry.