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

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 log resolution $\ov{X}\rar \cSU_C(2,\cO_C)$. Based on earlier work of M. Kerr and J. 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 the relative Chow group of null-homologous one cycles (tensored with $\Q$). This map takes values in the intermediate Jacobian of the compactly supported cohomology of the stable locus. We show that this is an isomorphism and since the intermediate Jacobian is identified with the Jacobian $Jac(C)\otimes \Q$, this can be thought of as a weak-representability result for open smooth varieties. A Hard Lefschetz theorem is also proved for the odd degree bottom weight cohomology of the moduli space $\cSU_C^s(2,\cO_C)$. When the rank $r\geq 2$, we compute the codimension two rational Chow groups of $\cSU_C(r,\cO_C)$.

💡 Research Summary

The paper investigates the geometry and Hodge‑theoretic properties of the moduli space
(\mathcal{SU}_C(r,\mathcal O_C)) of rank‑(r) semistable vector bundles with trivial determinant on a smooth projective curve (C) of genus (g). The authors focus primarily on the case (r=2), where F. Kirwan constructed a smooth log‑resolution (\overline{X}\to\mathcal{SU}_C(2,\mathcal O_C)) whose exceptional divisor (D) has normal‑crossing singularities.

Relative Chow groups and the Abel–Jacobi map.
Following the work of M. Kerr and J. Lewis, the paper adopts the notion of a relative Chow group (\CH_1(\overline{X},D;\mathbb Q)) with respect to the normal‑crossing divisor. Inside this group the authors isolate the homologically trivial part (\CH_1(\overline{X},D)_{\hom}). Using the machinery of mixed Hodge structures, they define an Abel–Jacobi map
\