Geodesic
On the standard conjecture for projective compactifications of N\'eron models of $3$-dimensional
Izvestiya. Mathematics , Tome 85 (2021) no. 1, pp. 145-175.

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We prove that the Grothendieck standard conjecture of Lefschetz type holds for a smooth complex projective $4$-dimensional variety $X$ fibred by Abelian varieties (possibly, with degeneracies) over a smooth projective curve if the endomorphism ring $\operatorname{End}_{\overline{\kappa(\eta)}} (X_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})$ 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 $X_\eta$ 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.
Keywords: standard conjecture, Abelian variety, Néron minimal model, toric rank. 
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S. G. Tankeev. On the standard conjecture for projective compactifications of N\'eron models of $3$-dimensional. Izvestiya. Mathematics , Tome 85 (2021) no. 1, pp. 145-175. http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a5/

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