On the standard conjecture of Lefschetz type for complex projective threefolds

摘要

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 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 : X ! C is birationally equivalent to either a ruled surface, an Enriques surface, or a K3-surface, 2) all the fibres of are smooth surfaces of Kodaira dimension { 6 0.