The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties

Abstract

We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on hyper-Kähler varieties of generalized Kummer type, and by constructing semiregular sheaves on abelian varieties. Our proof instead relies on a direct geometric relation between abelian fourfolds of Weil type with discriminant 1 and the six-dimensional hyper-Kähler varieties $\tilde{K}$ of O'Grady type arising as crepant resolutions $\tilde{K}\to K$ of a locally trivial deformation of a singular moduli space of sheaves on an abelian surface. As applications, we establish the Hodge conjecture and the Tate conjecture for any variety $\tilde{K}$ of OG6-type as above, and all of its powers.

final version, to appear in Journal de Mathématiques Pures et Appliquées