On an inductive approach to the standard conjecture for a fibred complex variety with strong semistable degeneracies

We prove that Grothendieck's standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the operators $\ast$ and $\Lambda$ of Hodge theory holds for a 4-dimensional smooth projective complex variety fibred over a smooth projective curve $C$ provided that every degenerate fibre is a union of smooth irreducible components of multiplicity 1 with normal crossings, the standard conjecture $B(X_{\overline\eta})$ holds for a generic geometric fibre $X_{\overline\eta}$, there is at least one degenerate fibre $X_\delta$ and the rational cohomology rings $H^\ast(V_i,\mathbb{Q})$ and $H^\ast(V_i\cap V_j,\mathbb{Q})$ of the irreducible components $V_i$ of every degenerate fibre $X_\delta=V_1+\dots+V_m$ are generated by classes of algebraic cycles. We obtain similar results for 3-dimensional fibred varieties with algebraic invariant cycles (defined by the smooth part $\pi'\colon X'\to C'$ of the structure morphism $\pi\colon X\to C$) or with a degenerate fibre all of whose irreducible components $E_i$ possess the property $H^2(E_i,\mathbb{Q})= \operatorname{NS}(E_i)\otimes_{\mathbb{Z}}\mathbb{Q}$.