On Abel-Jacobi maps of moduli of parabolic bundles over a curve

Let $C$ be a nonsingular complex projective curve, and $\mathcal{L}$ e a line bundle of degree 1 on $C$. Let $$\mathcal{M}$_{\alpha} := $\mathcal{M}$(r,$\mathcal{L}$,\alpha)$denotethemodulispaceof$S$-equivalence classes of Parabolic stable bundles of fixed rank $r$, determinant $\mathcal{L}$, full flags and generic weight $\alpha$. Let $n=$ dim${\mathcal{M}}_{\alpha }$. We aim to study the Abel-Jacobi maps for ${\mathcal{M}}_{\alpha }$ in the cases $k=2,n-1$. When $k=n-1$, we prove that the Abel-Jacobi map is a split surjection. When $k=2$ and $r=2$, we show that the Abel-Jacobi map is an isomorphism.