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

摘要

We prove that Grothendieck's standard conjecture B(X) of Lefschetz type on the algebraicity of the operators * 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-(eta) over bar) holds for a generic geometric fibre X-(eta) over bar, there is at least one degenerate fibre X-delta and the rational cohomology rings H*(V-i, Q) and H*(V-i boolean AND V-j, Q) of the irreducible components V-i of every degenerate fibre X-delta = V-1 + center dot center dot center dot + 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' : X' - C' of the structure morphism pi : X - C) or with a degenerate fibre all of whose irreducible components E-i possess the property H-2(E-i, Q) = NS(E-i) circle times(Z)Q.