The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties
Salvatore Floccari, Lie Fu
TL;DR
This work constructs a geometric bridge between abelian fourfolds of Weil type with discriminant $1$ and OG6-type hyper-Kähler varieties via singular OG6-varieties and their crepant resolutions. By proving the algebraicity of the Kuga--Satake correspondence for the associated K3 surfaces $S_K$, it deduces the Hodge conjecture for all powers of the Weil-type fourfolds and extends the KS--Hodge framework to OG6-resolutions, with Tate conjecture consequences under standard conjectures. The results are grounded in motivic relations showing that the key varieties are motivated by a common abelian fourfold $B_K$, and they yield broad consequences for the motives of hyper-Kähler varieties. Overall, the paper provides an independent, geometry-driven proof of major conjectures in this setting and expands the reach of Kuga--Satake techniques in the study of Weil-type and OG6-type objects.
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 $\widetilde{K}$ of O'Grady type arising as crepant resolutions $\widetilde{K}\to K$ 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 $\widetilde{K}$ of OG6-type as above, and all of its powers.