On the standard conjecture for projective compactifications of Néron models of -dimensional Abelian varieties

Abstract

Abstract 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.