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

TL;DR

This paper proves an Abel-Jacobi isomorphism for one cycles on Kirwan's resolution of the moduli space of rank 2 semistable bundles with trivial determinant, linking Chow groups, cohomology, and Jacobians.

Contribution

It establishes the Abel-Jacobi isomorphism for one cycles on Kirwan's resolution and computes Chow groups for higher ranks, advancing understanding of the geometry of these moduli spaces.

Findings

The Abel-Jacobi map is an isomorphism for null-homologous one cycles.

A Hard Lefschetz theorem is proved for certain cohomology groups.

Codimension two Chow groups are computed for higher rank moduli spaces.

Abstract

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 $\cSU_C(r,\cO_C)$.