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Izvestiya: Mathematics, 2003, Volume 67, Issue 3, Pages 597–635
DOI: https://doi.org/10.1070/IM2003v067n03ABEH000439
(Mi im439)
 

This article is cited in 14 scientific papers (total in 14 papers)

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

S. G. Tankeev

Vladimir State University
References:
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.
Received: 12.07.2001
Bibliographic databases:
UDC: 512.6
MSC: 14C25
Language: English
Original paper language: Russian
Citation: S. G. Tankeev, “On the standard conjecture for complex Abelian schemes over smooth projective curves”, Izv. Math., 67:3 (2003), 597–635
Citation in format AMSBIB
\Bibitem{Tan03}
\by S.~G.~Tankeev
\paper On the standard conjecture for complex Abelian schemes over smooth projective curves
\jour Izv. Math.
\yr 2003
\vol 67
\issue 3
\pages 597--635
\mathnet{http://mi.mathnet.ru//eng/im439}
\crossref{https://doi.org/10.1070/IM2003v067n03ABEH000439}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1992197}
\zmath{https://zbmath.org/?q=an:1072.14011}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33645642079}
Linking options:
  • https://www.mathnet.ru/eng/im439
  • https://doi.org/10.1070/IM2003v067n03ABEH000439
  • https://www.mathnet.ru/eng/im/v67/i3/p183
  • This publication is cited in the following 14 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
    Statistics & downloads:
    Abstract page:572
    Russian version PDF:290
    English version PDF:25
    References:67
    First page:1
     
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