We prove that the Grothendieck standard conjecture of Lefschetz type holds for a smooth complex projective -dimensional variety fibred by Abelian varieties (possibly, with degeneracies) over a smooth projective curve if the endomorphism ring of the generic geometric fibre is not an order of an imaginary quadratic field. This condition holds automatically in the cases when the reduction of the generic scheme fibre at some place of the curve is semistable in the sense of Grothendieck and has odd toric rank or the generic geometric fibre is not a simple Abelian variety.

中文翻译：

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关于维阿贝尔变种Nron模型的射影压缩的标准猜想

我们证明，如果通用几何纤维的内同态环不是虚构的阶，则Lefschetz类型的Grothendieck标准猜想适用于由Abelian品种（可能具有简并性）在光滑投影曲线上产生的光滑复投影维数品种二次域。如果在Grothendieck的意义上通用方案光纤在曲线的某个位置处的减小是半稳定的，并且具有奇数复曲面等级，或者通用几何光纤不是简单的Abelian品种，则这种情况自动成立。