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

We prove that Grothendieck's standard conjecture $B\left(X\right)$ 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 $B\left(X_{\overline{\eta }}\right)$ holds for a generic geometric fibre $X_{\overline{\eta }}$, there is at least one degenerate fibre $X_{\delta }$ and the rational cohomology rings $H^{\ast }(V_i,Q)$ and $H^{\ast }(V_i\cap V_j,Q)$ of the irreducible components $V_i$ of every degenerate fibre $X_{\delta }=V_1+\cdots +V_m$ 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 ${\pi }^{\prime }:X^{\prime }\to C^{\prime }$ of the structure morphism $\pi :X\to C$) or with a degenerate fibre all of whose irreducible components $E_i$ possess the property $H^2(E_i,Q)=NS\left(E_i\right){\otimes }_{Z}Q$. (paper)