Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal

TL;DR

This paper investigates the Abel-Jacobi map for smooth projective 3-folds, exploring conditions for surjectivity with rationally connected fibers, and relates these to the integral Hodge conjecture and decomposition of the diagonal.

Contribution

It establishes new links between the Abel-Jacobi map, rational connectedness, and the integral Hodge conjecture for specific classes of threefolds, including cubic threefolds.

Findings

Surjectivity of the Abel-Jacobi map with rationally connected fibers under certain conditions

Relation between rational connectedness and integral homological decomposition of the diagonal

Verification of the Hodge conjecture for degree 4 classes on fibrations into cubic threefolds

Abstract

Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Markoushevich. We then conclude that the Hodge conjecture holds for degree 4 integral Hodge classes on fibrations into cubic threefolds over curves, with restriction on singular fibers.