Geodesic
On the standard conjecture for a~$3$-dimensional variety fibred by curves with a~non-injective Kodaira--Spencer map
Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 1016-1035

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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 $1$ on some non-empty open subset of the surface. When the generic fibre of the structure morphism is of genus $2$, the condition on the endomorphisms of the Jacobian may be omitted.
Keywords: Grothendieck standard conjecture of Lefschetz type, Kodaira–Spencer map, Jacobian variety. 
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     author = {S. G. Tankeev},
     title = {On the standard conjecture for a~$3$-dimensional variety fibred by curves with a~non-injective {Kodaira--Spencer} map},
     journal = {Izvestiya. Mathematics },
     pages = {1016--1035},
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     number = {5},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2020_84_5_a8/}
}
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S. G. Tankeev. On the standard conjecture for a~$3$-dimensional variety fibred by curves with a~non-injective Kodaira--Spencer map. Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 1016-1035. http://geodesic.mathdoc.fr/item/IM2_2020_84_5_a8/