A remark on the Abel-Jacobi morphism for the cubic threefold

2012/12/30 by Ze Xu, Xu, Ze
Mathematics · #14C25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1212.6790

openalex publication_date 2012/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a smooth cubic threefold and J(X) be its intermediate Jacobian. We show that there exists a codimension 2 cycle Z on J(X)× X with Z_t homologically trivial for each t∈ J(X), such that the morphism ϕ_Z: J(X)→ J(X) induced by the Abel-Jacobi map is the identity. This answers positively a question of Voisin in the case of the cubic threefold.

Related

• Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal
• Cubic surfaces and cubic threefolds, jacobians and intermediate jacobians
• Abel-Jacobi maps associated to smooth cubic threefolds
• Extension of the Topological Abel-Jacobi Map for Cubic Threefolds
• The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14