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

We prove that Grothendieck's standard conjecture of Lefschetz type on the algebraicity of the operators and of Hodge theory holds for a 4-dimensional smooth projective complex variety fibred over a smooth projective curve provided that every degenerate fibre is a union of smooth irreducible components of multiplicity with normal crossings, the standard conjecture holds for a generic geometric fibre , there is at least one degenerate fibre and the rational cohomology rings and of the irreducible components of every degenerate fibre are generated by classes of algebraic cycles. We obtain similar results for -dimensional fibred varieties with algebraic invariant cycles (defined by the smooth part of the structure morphism ) or with a degenerate fibre all of whose irreducible components possess the property .