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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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