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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IM2_2021_85_1_a5,
     author = {S. G. Tankeev},
     title = {On the standard conjecture for projective compactifications of {N\'eron} models of $3$-dimensional},
     journal = {Izvestiya. Mathematics },
     pages = {145--175},
     publisher = {mathdoc},
     volume = {85},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a5/}
}
TY  - JOUR
AU  - S. G. Tankeev
TI  - On the standard conjecture for projective compactifications of N\'eron models of $3$-dimensional
JO  - Izvestiya. Mathematics 
PY  - 2021
SP  - 145
EP  - 175
VL  - 85
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a5/
LA  - en
ID  - IM2_2021_85_1_a5
ER  - 
%0 Journal Article
%A S. G. Tankeev
%T On the standard conjecture for projective compactifications of N\'eron models of $3$-dimensional
%J Izvestiya. Mathematics 
%D 2021
%P 145-175
%V 85
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a5/
%G en
%F IM2_2021_85_1_a5
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/