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 ##IMG## [http://ej.iop.org/images/1064-5632/84/5/1016/IZV_84_5_1016ieqn2.gif] {$1$} on some non-empty open subset of the surface. When the generic fibre of the structure morphism is of genus ##IMG## [http://ej.iop.org/images/1064-5632/84/5/1016/IZV_84_5_1016ieqn3.gif] {$2$} , the condition on the endomorphisms of the Jacobian may be omitted.

中文翻译：

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关于## IMG ##的标准猜想[http://ej.iop.org/images/1064-5632/84/5/1016/toc_IZV_84_5_1016ieqn1.gif] {$ 3 $}-由带有非内射式Kodaira–Spencer地图

我们证明了Lefschetz类型的Grothendieck标准猜想适用于由光滑纤维投射表面上的曲线（可能具有简并性）构成的复杂投射3维变体，前提是某些光滑纤维的Jacobian变体的内同形环与整数和相应的Kodaira–Spencer映射在某些非Google地图上的排名为## IMG ## [http://ej.iop.org/images/1064-5632/84/5/1016/IZV_84_5_1016ieqn2.gif] {$ 1 $}空的表面开放子集。当结构态态的类属纤维是属## IMG ## [http://ej.iop.org/images/1064-5632/84/5/1016/IZV_84_5_1016ieqn3.gif] {$ 2 $}时，条件关于雅可比行列式的内同态可以省略。