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Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 5, Pages 119–142 (Mi izv7826)  

This article is cited in 3 scientific papers (total in 3 papers)

On the standard conjecture for complex 4-dimensional elliptic varieties

S. G. Tankeev

Vladimir State University

Abstract: We prove that the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of operators $\ast$ and $\Lambda$ of Hodge theory holds for every smooth complex projective model $X$ of the fibre product $X_1\times_C X_2$, where $X_1\to C$ is an elliptic surface over a smooth projective curve $C$ and $X_2\to C$ is a morphism of a smooth projective threefold onto $C$ such that one of the following conditions holds: a generic geometric fibre $X_{2s}$ is an Enriques surface; all fibres of the morphism $X_2\to C$ are smooth $\mathrm{K}3$-surfaces and the Hodge group $\operatorname{Hg}(X_{2s})$ of the generic geometric fibre $X_{2s}$ has no geometric simple factors of type $A_1$ (the assumption on the Hodge group holds automatically if the number $22-\operatorname{rank}\operatorname{NS}(X_{2s})$ is not divisible by 4).

Keywords: elliptic variety, standard conjecture of Lefschetz type, Enriques surface, $\mathrm{K}3$-surface, Hodge group, algebraic cycle.

DOI: https://doi.org/10.4213/im7826

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English version:
Izvestiya: Mathematics, 2012, 76:5, 967–990

Bibliographic databases:

UDC: 512.6
MSC: 14C25, 14D07, 14F25, 14J35
Received: 08.08.2011

Citation: S. G. Tankeev, “On the standard conjecture for complex 4-dimensional elliptic varieties”, Izv. RAN. Ser. Mat., 76:5 (2012), 119–142; Izv. Math., 76:5 (2012), 967–990

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. G. Tankeev, “On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models”, Izv. Math., 78:1 (2014), 169–200  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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