On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models

We prove that the Grothendieck standard conjecture of Lefschetz type on the algebraicity of operators and of Hodge theory holds for every smooth complex projective model of the fibre product , where is an elliptic surface over a smooth projective curve and is a family of K3 surfaces with semistable degenerations of rational type such that for a generic geometric fibre . We also show that holds for any smooth projective compactification of the Neron minimal model of an Abelian scheme of relative dimension over an affine curve provided that the generic scheme fibre is an absolutely simple Abelian variety with reductions of multiplicative type at all infinite places.