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

We prove that the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of operators $*$ and $\Lambda$ of Hodge theory holds for every smooth complex projective model $X$ of the fibre product $X_1\times_CX_2$, where $X_1\to C$ is an elliptic surface over a smooth projective curve $C$ and $X_2\to C$ is a family of K3 surfaces with semistable degenerations of rational type such that $\operatorname{rank}\operatorname{NS}(X_{2s})\ne18$ for a generic geometric fibre $X_{2s}$. We also show that $B(X)$ holds for any smooth projective compactification $X$ of the Néron minimal model of an Abelian scheme of relative dimension $3$ 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.