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Izv. Akad. Nauk SSSR Ser. Mat., 1984, Volume 48, Issue 2, Pages 264–304 (Mi izv1446)  

This article is cited in 21 scientific papers (total in 21 papers)

Weights of simple Lie algebras in the cohomology of algebraic varieties

Yu. G. Zarhin


Abstract: This article studies representations of semisimple Lie algebras arising naturally in the $l$-adic cohomology of algebraic varieties defined over global fields. A conjecture is formulated about the restrictions the index of the cohomology space and the Hodge numbers of a variety impose on the weights of a represention. The conjecture is proved for ordinary varieties over function fields. An analog of the conjecture is valid for the rational cohomology of varieties defined over the field of complex numbers.
Bibliography: 41 titles.

Full text: PDF file (4260 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1985, 24:2, 245–281

Bibliographic databases:

UDC: 512.7
MSC: 17B10, 14J20, 14F30
Received: 22.03.1983

Citation: Yu. G. Zarhin, “Weights of simple Lie algebras in the cohomology of algebraic varieties”, Izv. Akad. Nauk SSSR Ser. Mat., 48:2 (1984), 264–304; Math. USSR-Izv., 24:2 (1985), 245–281

Citation in format AMSBIB
\Bibitem{Zar84}
\by Yu.~G.~Zarhin
\paper Weights of simple Lie algebras in the cohomology of algebraic varieties
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1984
\vol 48
\issue 2
\pages 264--304
\mathnet{http://mi.mathnet.ru/izv1446}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=740792}
\zmath{https://zbmath.org/?q=an:0579.14019}
\transl
\jour Math. USSR-Izv.
\yr 1985
\vol 24
\issue 2
\pages 245--281
\crossref{https://doi.org/10.1070/IM1985v024n02ABEH001230}


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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. Yu. G. Zarhin, “Representations of the Lie algebra $sl(2)$ in $l$-adic cohomologies”, Funct. Anal. Appl., 19:3 (1985), 217–218  mathnet  crossref  mathscinet  zmath  isi
    2. S. G. Tankeev, “Algebraic cycles on an abelian variety without complex multiplication”, Russian Acad. Sci. Izv. Math., 44:3 (1995), 531–553  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. S. G. Tankeev, “Cycles on Abelian varieties and exceptional numbers”, Izv. Math., 60:2 (1996), 391–424  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. S. G. Tankeev, “On Frobenius traces”, Izv. Math., 62:1 (1998), 157–190  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. S. G. Tankeev, “On weights of the $l$-adic representation and arithmetic of Frobenius eigenvalues”, Izv. Math., 63:1 (1999), 181–218  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. S. G. Tankeev, “Cycles of small codimension on a simple $2p$- or $4p$-dimensional Abelian variety”, Izv. Math., 63:6 (1999), 1221–1262  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. S. G. Tankeev, “The arithmetic and geometry of a generic hypersurface section”, Izv. Math., 66:2 (2002), 393–424  mathnet  crossref  crossref  mathscinet  zmath
    8. S. G. Tankeev, “On the standard conjecture for complex Abelian schemes over smooth projective curves”, Izv. Math., 67:3 (2003), 597–635  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. 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
    10. 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
    11. Argatov I.I., “Extension of an elastic space with a rigid bar”, Journal of Applied Mechanics and Technical Physics, 49:1 (2008), 98–104  crossref  isi  elib
    12. 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
    13. 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
    14. 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
    15. 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
    16. 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
    17. 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
    18. 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
    19. Orr M., “Lower Bounds For Ranks of Mumford-Tate Groups”, Bull. Soc. Math. Fr., 143:2 (2015), 229–246  isi
    20. 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
    21. 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 Академии наук СССР. Серия математическая Izvestiya: Mathematics
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