Toric geometry and the standard conjecture for a~compactification of the N\'eron model of Abelian variety

S. G. Tankeev

Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 140-171

Résumé

It is proved that if $\mathcal M\to C$ is the Néron minimal model of a principally polarized $(d-1)$-dimensional Abelian variety $\mathcal M_\eta$ over the field $\kappa(\eta)$ of rational functions of a smooth projective curve $C$, $$ \operatorname{End}_{\overline{\kappa(\eta)}} (\mathcal M_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})=\mathbb Z, $$ the complexification of the Lie algebra of the Hodge group $\operatorname{Hg}(M_\eta\otimes_{\kappa(\eta)}\mathbb {C})$ is a simple Lie algebra of type $C_{d-1}$, all bad reductions of the Abelian variety $\mathcal M_\eta$ are semi-stable, for any places $\delta,\delta'$ of bad reductions the $\mathbb Q$-space of Hodge cycles on the product $\operatorname{Alb}(\overline{\mathcal M_\delta^0})\,\times \, \operatorname{Alb}(\overline{\mathcal M_{\delta'}^0})$ of Albanese varieties is generated by classes of algebraic cycles, then there exists a finite ramified covering $\widetilde{C}\to C$ such that, for any Künnemann compactification $\widetilde{X}$ of the Néron minimal model of the Abelian variety $\mathcal M_\eta\otimes_{\kappa(\eta)}\kappa(\widetilde{\eta})$, the Grothendieck standard conjecture $B(\widetilde{X})$ of Lefschetz type is true.

Détail

Keywords: toric geometry, Grothendieck standard conjecture of Lefschetz type, Abelian variety, Künnemann compactification of Néron model, Hodge conjecture.