Toric geometry and the standard conjecture for a compactification of the Neron model of Abelian varietyover $1$-dimensional function field

It is proved that if$\mathcal M\to C$ is the Neron 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 reductionsthe $\mathbb Q$-space of Hodge cycles on the product$\operatorname{Alb}(\overline{\mathcal M_\delta^0}) \times   \operatorname{Alb}(\overline{\mathcal M_{\delta'}^0})$ of Albanese varietiesis generated by classes of algebraic cycles,thenthere exists a finite ramified covering $\widetilde{C}\to C$ such that, for any Künnemann compactification $\widetilde{X}$of the Neron 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.