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 Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 1, Pages 181–214 (Mi izv8041)

On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models

S. G. Tankeev

Abstract: We prove that the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of operators $*$ and $\Lambda$ of Hodge theory holds for every smooth complex projective model $X$ of the fibre product $X_1\times_CX_2$, where $X_1\to C$ is an elliptic surface over a smooth projective curve $C$ and $X_2\to C$ is a family of K3 surfaces with semistable degenerations of rational type such that $\operatorname{rank}\operatorname{NS}(X_{2s})\ne18$ for a generic geometric fibre $X_{2s}$. We also show that $B(X)$ holds for any smooth projective compactification $X$ of the Néron minimal model of an Abelian scheme of relative dimension $3$ over an affine curve provided that the generic scheme fibre is an absolutely simple Abelian variety with reductions of multiplicative type at all infinite places.

Keywords: elliptic variety, standard conjecture of Lefschetz type, K3 surface, semistable degeneration of rational type, algebraic cycle, Néron minimal model, reduction of multiplicative type.

 Funding Agency Grant Number Russian Foundation for Basic Research 12-01-00097

DOI: https://doi.org/10.4213/im8041

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English version:
Izvestiya: Mathematics, 2014, 78:1, 169–200

Bibliographic databases:

UDC: 512.6
MSC: 14C25, 14D07, 14F25, 14J35

Citation: S. G. Tankeev, “On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models”, Izv. RAN. Ser. Mat., 78:1 (2014), 181–214; Izv. Math., 78:1 (2014), 169–200

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8041
• https://doi.org/10.4213/im8041
• http://mi.mathnet.ru/eng/izv/v78/i1/p181

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This publication is cited in the following articles:
1. O. V. Nikol'skaya, “On Algebraic Cohomology Classes on a Smooth Model of a Fiber Product of Families of K3 surfaces”, Math. Notes, 96:5 (2014), 745–752
2. S. G. Tankeev, “On the standard conjecture and the existence of a Chow–Lefschetz decomposition for complex projective varieties”, Izv. Math., 79:1 (2015), 177–207
3. O. V. Nikolskaya, “Ob algebraicheskikh tsiklakh na rassloennykh proizvedeniyakh neizotrivialnykh semeistv regulyarnykh poverkhnostei s geometricheskim rodom 1”, Model. i analiz inform. sistem, 23:4 (2016), 440–465
4. S. G. Tankeev, “On an inductive approach to the standard conjecture for a fibred complex variety with strong semistable degeneracies”, Izv. Math., 81:6 (2017), 1253–1285
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