The Abel–Jacobi isomorphism for one-cycles on Kirwan's log resolution of the moduli space 𝒮𝒰 C (2,𝒪 C )

Abstract

Abstract In this paper, we consider the moduli space 𝒮 𝒰 C ( r , 𝒪 C ) $\mathcal {S\hspace{-1.0pt}U}_C(r,\mathcal {O}_C)$ of rank r semistable vector bundles with trivial determinant on a smooth projective curve C of genus g. For r = 2, F. Kirwan constructed a smooth log resolution X ¯ → 𝒮 𝒰 C ( 2 , 𝒪 C ) $\overline{X}\rightarrow \mathcal {S\hspace{-1.0pt}U}_C(2,\mathcal {O}_C)$ . Based on earlier work of M. Kerr and J. Lewis, Lewis explains in the Appendix the notion of a relative Chow group (w.r.t. 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 ) ⊗ ℚ $\operatorname{Jac}(C)\otimes \mathbb {Q}$ , 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 𝒮 𝒰 C s ( 2 , 𝒪 C ) $\mathcal {S\hspace{-1.0pt}U}_C^s(2,\mathcal {O}_C)$ . When r ≥ 2, we compute the codimension two rational Chow groups of 𝒮 𝒰 C ( r , 𝒪 C ) $\mathcal {S\hspace{-1.0pt}U}_C(r,\mathcal {O}_C)$ .