On algebraic isomorphisms of rational cohomology of a Künneman compactification of the Néron minimal model

It is proved that the Grothendieck standard conjecture of Lefschetz type holds for rational cohomology of degree 2 or 3 of a Künnemann compactification of the Néron minimal model of an absolutely simple principally polarized Abelian variety over the field of rational functions of a smooth projective curve under certain restrictions on the ring of endomorphisms of the Abelian variety.