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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 1, Pages 175–196 (Mi izv2744)  

This article is cited in 10 scientific papers (total in 10 papers)

On the standard conjecture of Lefschetz type for complex projective threefolds

S. G. Tankeev

Vladimir State University

Abstract: Under certain natural assumptions on cohomology of a complex projective fibred threefold with semi-stable degenerations, we prove the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the operators $\Lambda$ and $*$. In particular, $B(X)$ is true if at least one of the following conditions holds: 1) the generic fibre of some $1$-parameter holomorphic family $\pi\colon X\to C$ is birationally equivalent to either a ruled surface, an Enriques surface, or a K3-surface, 2) all the fibres of $\pi$ are smooth surfaces of Kodaira dimension $\varkappa\le0$.

Keywords: standard conjecture of Lefschetz type.

DOI: https://doi.org/10.4213/im2744

Full text: PDF file (609 kB)
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English version:
Izvestiya: Mathematics, 2010, 74:1, 167–187

Bibliographic databases:

UDC: 512.6
MSC: 14C25, 14F25, 14J30
Received: 01.11.2007

Citation: S. G. Tankeev, “On the standard conjecture of Lefschetz type for complex projective threefolds”, Izv. RAN. Ser. Mat., 74:1 (2010), 175–196; Izv. Math., 74:1 (2010), 167–187

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. S. G. Tankeev, “On the standard conjecture of Lefschetz type for complex projective threefolds. II”, Izv. Math., 75:5 (2011), 1047–1062  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. S. G. Tankeev, “On the standard conjecture for complex 4-dimensional elliptic varieties”, Izv. Math., 76:5 (2012), 967–990  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. O. V. Nikol'skaya, “On algebraic cycles on a fibre product of families of K3-surfaces”, Izv. Math., 77:1 (2013), 143–162  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Gorchinskiy S., Guletskii V., “Non-Trivial Elements in the Abel-Jacobi Kernels of Higher-Dimensional Varieties”, Adv. Math., 241 (2013), 162–191  crossref  mathscinet  zmath  isi  elib  scopus
    5. 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
    6. 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
    7. 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  mathnet  crossref  mathscinet  elib
    8. 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  mathnet  crossref  crossref  adsnasa  isi  elib
    9. S. G. Tankeev, “O standartnoi gipoteze dlya rassloennogo nad poverkhnostyu 3-mernogo mnogoobraziya”, Matem. zametki, 105:4 (2019), 643–644  mathnet  crossref  elib
    10. S. G. Tankeev, “O standartnoi gipoteze dlya rassloennogo proizvedeniya trekh ellipticheskikh poverkhnostei s poparno neperesekayuschimisya diskriminantnymi lokusami”, Izv. RAN. Ser. matem., 83:3 (2019), 213–256  mathnet  crossref
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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