On the standard conjecture and the existence of a Chow–Lefschetz decomposition for complex projective varieties

We prove the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the operators $*$ and $\Lambda$ of Hodge theory for a smooth complex projective variety $X$ if at least one of the following conditions holds: $X$ is a compactification of the Néron minimal model of an Abelian scheme of relative dimension $3$ over an affine curve, and the generic scheme fibre of the Abelian scheme has reductions of multiplicative type at all infinite places; $X$ is an irreducible holomorphic symplectic (hyperkähler) 4-dimensional variety that coincides with the Altman–Kleiman compactification of the relative Jacobian variety of a family $\mathcal C\to\mathbb P^2$ of hyperelliptic curves of genus 2 with weak degenerations, and the canonical projection $X\to\mathbb P^2$ is a Lagrangian fibration. We also show that a Chow–Lefschetz decomposition exists for every smooth projective 3-dimensional variety $X$ which has the structure of a 1-parameter non-isotrivial family of K3-surfaces (with degenerations) or a family of regular surfaces of arbitrary Kodaira dimension $\varkappa$ with strong degenerations.