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 $\mathcal{M}_{\alpha} := \mathcal{M}(r,\mathcal{L},\alpha)$ denote the moduli space of $S$-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.

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