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This article is cited in 6 scientific papers (total in 6 papers)
On the standard conjecture and the existence of a Chow–Lefschetz decomposition for complex projective varieties
S. G. Tankeev Vladimir State University
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.
DOI:
https://doi.org/10.4213/im8227
Full text:
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English version:
Izvestiya: Mathematics, 2015, 79:1, 177–207
Bibliographic databases:
UDC:
512.7
MSC: 14C25, 14F25, 14J30, 14J35 Received: 28.02.2014
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
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Linking options:
http://mi.mathnet.ru/eng/izv8227https://doi.org/10.4213/im8227 http://mi.mathnet.ru/eng/izv/v79/i1/p185
Citing articles on Google Scholar:
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This publication is cited in the following articles:
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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
-
J. Suh, “Standard conjecture of Künneth type with torsion coefficients”, Algebra Number Theory, 11:7 (2017), 1573–1596
-
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
-
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
-
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
-
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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