On the standard conjecture of Lefschetz type for complex projective threefolds

Under certain natural assumptions on cohomology of a complex projective fibred threefold with semi-stable degenerations, we prove the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the operators $\Lambda$ and $*$. In particular, $B(X)$ is true if at least one of the following conditions holds: 1) the generic fibre of some $1$-parameter holomorphic family $\pi\colon X\to C$ is birationally equivalent to either a ruled surface, an Enriques surface, or a K3-surface, 2) all the fibres of $\pi$ are smooth surfaces of Kodaira dimension $\varkappa\le0$.