On the standard conjecture for a -dimensional variety fibred by curves with a non-injective Kodaira–Spencer map

Abstract

Abstract We prove that the Grothendieck standard conjecture of Lefschetz type holds for a complex projective 3-dimensional variety fibred by curves (possibly with degeneracies) over a smooth projective surface provided that the endomorphism ring of the Jacobian variety of some smooth fibre coincides with the ring of integers and the corresponding Kodaira–Spencer map has rank on some non-empty open subset of the surface. When the generic fibre of the structure morphism is of genus , the condition on the endomorphisms of the Jacobian may be omitted.