Stability of the parabolic Picard sheaf

Let X be a smooth irreducible complex projective curve of genus g ≥ 2 , and let D = x 1 + ⋯ + x r be a reduced effective divisor on X . Denote by U α ( L ) the moduli space of stable parabolic vector bundles on X of rank n , determinant L of degree d with flag type { { k j i } j = 1 m i } i = 1 r . Assume that the greatest common divisor of the collection of integers { degree ( L ) , { k j i } j = 1 m i } i = 1 r } } is 1; this condition ensures that there is a Poincaré parabolic vector bundle on X × U α ( L ) . The direct image, to U α ( L ) , of the vector bundle underlying the Poincaré parabolic vector bundle is called the parabolic Picard sheaf. We prove that the parabolic Picard sheaf is stable.