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Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 3, Pages 183–224 (Mi izv439)  

This article is cited in 8 scientific papers (total in 8 papers)

On the standard conjecture for complex Abelian schemes over smooth projective curves

S. G. Tankeev

Vladimir State University

Abstract: We reduce the Hodge conjecture for Abelian varieties to the question of the existence of an algebraic isomorphism $H^2(C,R^{2d-i}\pi_\ast\mathbb Q)\widetilde\rightarrow, H^0(C,R^i\pi_\ast\mathbb Q)$ for all $i\geqslant 2$ and all principally polarized complex Abelian schemes $\pi\colon X\to C$ of relative dimension $d$ over smooth projective curves. If the canonically defined Hodge cycles $\alpha_i(X/C)\in H^0(C,R^i\pi_\ast\mathbb Q)\otimes H^0(C,R^i\pi_\ast\mathbb Q)$ are algebraic for all integers $i\geqslant 2$, then the Grothendieck standard conjecture $B(X)$ on the algebraicity of the operators $\Lambda$ and $\ast$ holds for $X$. We prove $B(X)$ for an Abelian scheme under the assumption that $\operatorname{End}(X_s)=\mathbb Z$ for some geometric fibre $X_s$ of non-exceptional dimension.

DOI: https://doi.org/10.4213/im439

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English version:
Izvestiya: Mathematics, 2003, 67:3, 597–635

Bibliographic databases:

UDC: 512.6
MSC: 14C25
Received: 12.07.2001

Citation: S. G. Tankeev, “On the standard conjecture for complex Abelian schemes over smooth projective curves”, Izv. RAN. Ser. Mat., 67:3 (2003), 183–224; Izv. Math., 67:3 (2003), 597–635

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 numerical equivalence of algebraic cycles on potentially simple Abelian schemes of prime relative dimension”, Izv. Math., 69:1 (2005), 143–162  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. S. G. Tankeev, “Monoidal transformations and conjectures on algebraic cycles”, Izv. Math., 71:3 (2007), 629–655  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. S. G. Tankeev, “On algebraic cycles on complex Abelian schemes over smooth projective curves”, Izv. Math., 72:4 (2008), 817–844  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. 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
    5. 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
    6. 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
    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, “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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