## On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models.(English. Russian original)Zbl 1290.14007

Izv. Math. 78, No. 1, 169-200 (2014); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 1, 181-214 (2014).
Let $$X$$ be an arbitrary smooth complex projective model of the fibre product $$X_1\times_C X_2$$, where $$X_1\rightarrow C$$ is an elliptic surface over a smooth projective curve $$C$$ and $$X_2\rightarrow C$$ is a family of $$K3$$ surfaces. Such a family is said to be with semistable degenerations of rational type, if, for every singular fibre, the Picard-Lefschetz transformation $$\gamma$$ satisfies the condition $$(\log(\gamma))^2\neq 0$$. In the present paper the author proves that the Grothendieck standard conjecture $$B(X)$$ of Lefschetz type holds for $$X$$ under the assumption that $$X_2\rightarrow C$$ is a family of $$K3$$ surfaces with semistable degenerations of rational type such that rank NS$$(X_{2s})\neq 18$$ for a generic geometric fibre $$X_{2s}$$. Furthermore he proves that $$B(X)$$ holds for every smooth projective compactification $$X$$ of the Néron minimal model of an abelian scheme of relative dimension 3 over an affine curve if the generic fibre is an absolutely simple abelian variety with reductions of multiplicative type at all infinite places.

### MSC:

 14C25 Algebraic cycles 14D07 Variation of Hodge structures (algebro-geometric aspects) 14F25 Classical real and complex (co)homology in algebraic geometry 14J35 $$4$$-folds
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