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Izv. RAN. Ser. Mat., 2007, Volume 71, Issue 3, Pages 197–224 (Mi izv550)  

This article is cited in 9 scientific papers (total in 9 papers)

Monoidal transformations and conjectures on algebraic cycles

S. G. Tankeev

Vladimir State University

Abstract: We consider the following conjectures: $\operatorname{Hodge}(X)$, $\operatorname{Tate}(X)$ (over a perfect finitely generated field), Grothendieck's standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the Hodge operator $\ast$, conjecture $D(X)$ on the coincidence of the numerical and homological equivalences of algebraic cycles and conjecture $C(X)$ on the algebraicity of Künneth components of the diagonal for smooth complex projective varieties. We show that they are compatible with monoidal transformations: if one of them holds for a smooth projective variety $X$ and a smooth closed subvariety $Y\hookrightarrow X$, then it holds for $X'$, where $f\colon X'\to X$ is the blow up of $X$ along $Y$. All of these conjectures are reduced to the case of rational varieties.

DOI: https://doi.org/10.4213/im550

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English version:
Izvestiya: Mathematics, 2007, 71:3, 629–655

Bibliographic databases:

UDC: 512.6
MSC: 14C25, 14F99
Received: 05.10.2004

Citation: S. G. Tankeev, “Monoidal transformations and conjectures on algebraic cycles”, Izv. RAN. Ser. Mat., 71:3 (2007), 197–224; Izv. Math., 71:3 (2007), 629–655

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 of Lefschetz type for complex projective threefolds”, Izv. Math., 74:1 (2010), 167–187  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. 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
    3. 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
    4. 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
    5. O. V. Nikol'skaya, “On the geometry of a smooth model of a fibre product of families of K3 surfaces”, Sb. Math., 205:2 (2014), 269–276  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. 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
    7. 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
    8. Robert Laterveer, “Yet another version of Mumford’s theorem”, Arch. Math, 2015  crossref  mathscinet  scopus
    9. 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
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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