Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class

Bert van Geemen; Antonio Rapagnetta

Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class

Algebraic Geometry 2026-07-20 v1

Authors: Bert van Geemen , Antonio Rapagnetta

Abstract

There are now several proofs of the Hodge conjecture for the general abelian fourfold of Weil type with trivial discriminant. This paper provides another one. The abelian fourfolds under consideration allow a map to a hyperk\"ahler sixfold of K3 $^{[3]}$ type. The pull-back of the second Chern class of the tangent bundle of the sixfold is an algebraic class in codimension two that is not an intersection of divisor classes and the main result follows. After recalling the basic facts on abelian fourfolds of Weil type we establish the existence of the map using results on the birational geometry of these hyperk\"ahler manifolds and deformation theory.

Keywords

Hodge conjecture abelian fourfold Weil type hyperkähler sixfold algebraic class deformation theory

Cite

@article{arxiv.2607.18341,
  title  = {Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class},
  author = {Bert van Geemen and Antonio Rapagnetta},
  journal= {arXiv preprint arXiv:2607.18341},
  year   = {2026}
}