The Abel-Jacobi isomorphism on one cycles on the moduli space of vector bundles with trivial determinant on a curve

TL;DR

This paper proves that the Abel-Jacobi map on the rational Chow group of one cycles on the moduli space of stable vector bundles with trivial determinant on a curve is an isomorphism onto the Jacobian of the curve, for certain ranks and genera.

Contribution

It establishes the Abel-Jacobi isomorphism for one cycles on the moduli space of vector bundles with trivial determinant, linking algebraic cycles to the Jacobian.

Findings

The Abel-Jacobi map is an isomorphism for the specified moduli space.

The isomorphism connects the Chow group to the Jacobian of the curve.

The result applies for rank r ≥ 2 and genus g ≥ 4.

Abstract

We consider the moduli space $\cSU_C^s(r,\cO_C)$ of rank r stable vector bundles with trivial determinant on a smooth projective curve $C$ of genus $g$. We show that the Abel-Jacobi map on the rational Chow group $CH_1(\cSU_C^s(r,\cO_C))_{hom}\otimes \Q$ of one cycles which are homologous to zero, is an isomorphism onto the bottom weight intermediate Jacobian, which is identified with the Jacobian $Jac(C)\otimes \Q$. The result holds whenever $r\geq 2$ and $g\geq 4$.