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 Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 1, Pages 185–216 (Mi izv8227)

On the standard conjecture and the existence of a Chow–Lefschetz decomposition for complex projective varieties

S. G. Tankeev

Abstract: We prove the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the operators $*$ and $\Lambda$ of Hodge theory for a smooth complex projective variety $X$ if at least one of the following conditions holds: $X$ is a compactification of the Néron minimal model of an Abelian scheme of relative dimension $3$ over an affine curve, and the generic scheme fibre of the Abelian scheme has reductions of multiplicative type at all infinite places; $X$ is an irreducible holomorphic symplectic (hyperkähler) 4-dimensional variety that coincides with the Altman–Kleiman compactification of the relative Jacobian variety of a family $\mathcal C\to\mathbb P^2$ of hyperelliptic curves of genus 2 with weak degenerations, and the canonical projection $X\to\mathbb P^2$ is a Lagrangian fibration. We also show that a Chow–Lefschetz decomposition exists for every smooth projective 3-dimensional variety $X$ which has the structure of a 1-parameter non-isotrivial family of K3-surfaces (with degenerations) or a family of regular surfaces of arbitrary Kodaira dimension $\varkappa$ with strong degenerations.

Keywords: standard conjecture of Lefschetz type, Néron minimal model, reduction of multiplicative type, K3-surface, hyperkähler variety, Chow–Lefschetz decomposition, Abel–Jacobi map.

 Funding Agency Grant Number Russian Foundation for Basic Research 12-01-00097 This paper was written with the financial support of RFBR (grant no. 12-01-00097).

DOI: https://doi.org/10.4213/im8227

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English version:
Izvestiya: Mathematics, 2015, 79:1, 177–207

Bibliographic databases:

UDC: 512.7
MSC: 14C25, 14F25, 14J30, 14J35

Citation: S. G. Tankeev, “On the standard conjecture and the existence of a Chow–Lefschetz decomposition for complex projective varieties”, Izv. RAN. Ser. Mat., 79:1 (2015), 185–216; Izv. Math., 79:1 (2015), 177–207

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8227
• https://doi.org/10.4213/im8227
• http://mi.mathnet.ru/eng/izv/v79/i1/p185

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. 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
2. J. Suh, “Standard conjecture of Künneth type with torsion coefficients”, Algebra Number Theory, 11:7 (2017), 1573–1596
3. S. G. Tankeev, “On the standard conjecture for a fibre product of three elliptic surfaces with pairwise-disjoint discriminant loci”, Izv. Math., 83:3 (2019), 613–653
4. S. G. Tankeev, “On algebraic isomorphisms of rational cohomology of a Künneman compactification of the Néron minimal model”, Sib. elektron. matem. izv., 17 (2020), 89–125
5. S. G. Tankeev, “On the standard conjecture for a $3$-dimensional variety fibred by curves with a non-injective Kodaira–Spencer map”, Izv. Math., 84:5 (2020), 1016–1035
6. S. G. Tankeev, “On the standard conjecture for projective compactifications of Néron models of $3$-dimensional Abelian varieties”, Izv. Math., 85:1 (2021), 145–175
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