Stability of the parabolic Picard sheaf

Let X be a smooth irreducible complex projective curve of genus \(g\,\ge \, 2\) $g\ge 2$ , and let \(D\,=\,x_1+\dots +x_r\) $D=x_1+\cdots +x_r$ be a reduced effective divisor on X. Denote by \(U_{\alpha }(L)\) $U_{\alpha }(L)$ the moduli space of stable parabolic vector bundles on X of rank n, determinant L of degree d with flag type \(\{\{k^i_j\}_{j=1}^{m_i}\}_{i=1}^r\) ${\{{\{k_j^i\}}_{j=1}^{m_i}\}}_{i=1}^r$ . Assume that the greatest common divisor of the collection of integers \(\{\text {degree}(L),\, \{k^i_j\}_{j=1}^{m_i}\}_{i=1}^r\}\}\) ${\{\text{degree}(L),{\{k_j^i\}}_{j=1}^{m_i}\}}_{i=1}^r\}\}$ is 1; this condition ensures that there is a Poincaré parabolic vector bundle on \(X\times U_{\alpha }(L)\) $X\times U_{\alpha }(L)$ . The direct image, to \(U_{\alpha }(L)\) $U_{\alpha }(L)$ , 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.