Non-trivial elements in the Abel-Jacobi kernels of higher-dimensional varieties (Q390998)

Let \(X\) be a smooth projective variety. The Bloch-Belinson conjecture says that, if \(X\) is defined over a number field, then the Abel-Jacobi map NEWLINE\[NEWLINE\text{CH}^p(X;{\mathbb Q})_0 \longrightarrow J^p(X_{\mathbb C})\otimes {\mathbb Q}NEWLINE\]NEWLINE is injective, where \(\text{CH}^p(X; {\mathbb Q})_0\) is the subgroup in \(\text{CH}^p(X; {\mathbb Q})\) of cycles homologically equivalent to zero, and \(J^p(X_{\mathbb C})\) is the \(p\)-th intermediate Jacobian. Therefore non-trivial algebraic cycles in the Abel-Jacobi kernel are expected to be rational over an extension of \({\mathbb Q}\) whose transcendence degree is at least one.NEWLINENEWLINEThe first example of such an algebraic cycle was constructed on the product of two curves by \textit{C. Schoen} [Proc. Symp. Pure Math. 46, 463--473 (1987; Zbl 0647.14002)].NEWLINENEWLINEIn this paper, the authors present a technique to construct non-trivial elements in the kernel of the Abel-Jacobi map in any codimension. The main result is the following:NEWLINENEWLINETheorem. Let \(X\) and \(S\) be two irreducible smooth varieties over an algebraically closed subfield in \({\mathbb C}\) and let \(d=\dim S\). Let \(\alpha\) be an element in \(\text{CH}^p(S \times X;{\mathbb Q})\). Suppose that there exists \(i\leq d-2\), such that the induced map NEWLINE\[NEWLINE({\alpha}_{\mathbb C})_*:H^{i,d}(S_{{\mathbb C}}) \rightarrow H^{i+p-d,p}(X_{\mathbb C})NEWLINE\]NEWLINE is non-zero. Let \(x\) be an arbitrary close point on \(S\). Then the difference \(\alpha'(x)\) between specialisations of \(\alpha\) (modified by the Albanese projector) at the generic point and at \(x\) is always a non-trivial element in the Abel-Jacobi kernel of the variety \(X_{\mathbb C}\).NEWLINENEWLINESpecialization arguments are an essential tool to prove that \(\alpha'(x)\) is in the Abel-Jacobi kernel. In order to prove the non-triviality of \(\alpha'(x)\), the authors show, that it follows from the condition on the induced map \(({\alpha}_{\mathbb C})_*\) in the statement of the main theorem.NEWLINENEWLINEThe authors also demonstrate how the main result works in practice: they apply it to \(K3\)-surfaces and to certain three-folds.