Syzygies and the Abel-Jacobi map for cyclic coverings (Q1325193)

Let \(Z\) be a threefold: there exist two Koszul type complexes \(K^ 2\), \(K^ 3\) of cohomology groups of \(Z\) such that if \(K^ 2\), \(K^ 3\) are exact, then the Abel Jacobi map for 2-dimensional cycles of \(Z\) has torsion image. If \(Z\) is a cyclic cover of a threefold \(X\), then the Galois group of the covering map \(Z\to X\) acts naturally on these complexes, which therefore split as sums of complexes of cohomology groups on \(X\), indexed by the characters of the Galois group. So exactness can be proven by showing exactness of certain complexes on \(X\). This way, the author proves: (1) If \(X = \mathbb{P}^ 3\) and \(Z\) is a cyclic cover of degree \(N\) of \(X\), branched on a general surface of degree \(Nd\), with \(d \geq 5\), then the Abel-Jacobi map of \(Z\) has torsion image. (2) If \(X \neq \mathbb{P}^ 3\) and \(Z \to X\) is a cyclic covering branched on a sufficiently ample divisor (the notion of sufficiently ample is made precise in the paper), generic in its linear system, then the image of the Abel-Jacobi map is contained in the pull-back of the intermediate Jacobian of \(X\).