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In this paper, we consider the moduli space $\cSU_C(r,\cO_C)$ of rank $r$semistable vector bundles with trivial determinant on a smooth projective curve$C$ of genus $g$. When the rank $r=2$, F. Kirwan constructed a smooth logresolution $\ov{X}\rar \cSU_C(2,\cO_C)$. Based on earlier work of M. Kerr andJ. Lewis, Lewis explains in the Appendix the notion of a relative Chow group(w.r.to the normal crossing divisor), and a subsequent Abel-Jacobi map on therelative Chow group of null-homologous one cycles (tensored with $\Q$). Thismap takes values in the intermediate Jacobian of the compactly supportedcohomology of the stable locus. We show that this is an isomorphism and sincethe intermediate Jacobian is identified with the Jacobian $Jac(C)\otimes \Q$,this can be thought of as a weak-representability result for open smoothvarieties. A Hard Lefschetz theorem is also proved for the odd degree bottomweight cohomology of the moduli space $\cSU_C^s(2,\cO_C)$. When the rank $r\geq2$, we compute the codimension two rational Chow groups of $\cSU_C(r,\cO_C)$.

Abel-Jacobi isomorphism for one cycles on Kirwan's resolution of the moduli space SU_C(2,O_C)