Geodesic
On the standard conjecture of Lefschetz type for complex projective threefolds
Izvestiya. Mathematics , Tome 74 (2010) no. 1, pp. 167-187

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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 $\Lambda$ 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 $\pi\colon X\to C$ is birationally equivalent to either a ruled surface, an Enriques surface, or a K3-surface, 2) all the fibres of $\pi$ are smooth surfaces of Kodaira dimension $\varkappa\le0$.
Mots-clés : standard conjecture of Lefschetz type. 
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     author = {S. G. Tankeev},
     title = {On the standard conjecture of {Lefschetz} type for complex projective threefolds},
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S. G. Tankeev. On the standard conjecture of Lefschetz type for complex projective threefolds. Izvestiya. Mathematics , Tome 74 (2010) no. 1, pp. 167-187. http://geodesic.mathdoc.fr/item/IM2_2010_74_1_a3/