Algebraic cycles on Prym varieties (Q431242)

Let \(X\) be an abelian variety over \(\mathbb{C}\), and let \(V\subset X\) be a subvariety. For any \(k\in\mathbb{Z}\) the Beauville grading \((s)\) is characterized by the property that \(x\in\text{CH}^p(X)_{(s)}\) if and only if \(k^*x=k^{2p-s}x\). In this paper the author proposes the following generalization of tautological rings: The small (resp. big) tautological ring \(\text{taut}(X,V)\) (resp. \(\text{Taut}(X,V)\)) of the pair \((X,V)\) is the smallest subring of \(\text{CH}(X)\) under the intersection product, which contain \(V\) (resp. \(\text{CH}(V)\)) and are stable under the Pontryagin product, the Fourier transform, \(k^*\), and \(k_*\) for all \(k\in\mathbb{Z}\). Furthermore the image of \(\text{Taut}(X,V)\) in the quotient \(A(X)\) of the Chow ring modulo algebraic equivalence is called the tautological ring of \((X,V)\). By applying this notion to the investigation of the Chow ring of Prym varieties, he shows that, under certain hypotheses, the special subvarieties of Prym varieties are algebraically equivalent and their classes belong to the tautological ring of the pair \((P,\psi (\tilde{C}))\), where \(P\) denotes the Prym variety associated to a degree two morphism \(\tilde{C}\rightarrow C\), which is either étale or ramified at two points, and \(\psi : \tilde{C}\rightarrow P\) is the Abel-Prym map.