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.


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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