A correspondence between the moduli spaces of vector bundles over a curve
The author establishes a correspondence between a smooth moduli space of vector bundles over a curve and a self-product of the Jacobian, and proves the nontriviality of the Griffiths group of the moduli space for a general curve. Let \(X\) be a smooth algebraic curve over \(\mathbb{C}\), and \(r\) and \(d\) be two coprime positive integers. Let \(\mathcal N\) be the moduli space of isomorphism classes of stable rank-\(r\) vector bundles over \(X\) with a fixed determinant of degree \(d\). Then, it is well-known that \(\mathcal N\) is smooth and there exists a universal bundle \(\mathcal E\) over \(X \times \mathcal N\). The characteristic classes \(a_k \in H^{2k}(X \times \mathcal N, \mathbb{Q})\) (\(2 \leq k \leq r\)) of \(\mathbb{P}(\mathcal E)\) give rise to homomorphisms from \(H_1(X, \mathbb{Q})\) to \(H^{2k-1}(\mathcal N, \mathbb{Q})\) by the slant product operation. These homomorphisms induce an algebra homomorphism from \(H_*(J^{r-1}, \mathbb{Q})\) to \(H^*(\mathcal N, \mathbb{Q})\), where \(J^{r-1}\) is the \((r-1)\)-fold product of the Jacobian \(J\). The main result of the paper says that the above algebra homomorphism is induced by an algebraic cycle on the product \(\mathcal N \times J^{r-1}\). Moreover, this algebraic cycle is canonical as an element of the Chow ring (cycles modulo rational equivalence) of \(\mathcal N \times J^{r-1}\). As an application of the main result, the author constructs non-zero elements in the Griffiths group of the moduli space \(\mathcal N\) for a general curve \(X\). These elements are obtained from the non-zero elements of the Jacobian \(J\) of a general curve discovered by G. Ceresa.
scientific article; zbMATH DE number 1347786
scientific article