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Izv. RAN. Ser. Mat., 2013, Volume 77, Issue 1, Pages 145–164 (Mi izv7939)  

This article is cited in 7 scientific papers (total in 7 papers)

On algebraic cycles on a fibre product of families of K3-surfaces

O. V. Nikol'skaya

Vladimir State University

Abstract: We prove the Hodge conjecture and the standard conjecture of Lefschetz type for fibre squares of smooth projective non-isotrivial families of $\mathrm K3$-surfaces over a smooth projective curve under the assumption that the rank of the lattice of transcendental cycles on a generic geometric fibre of the family is an odd prime. We prove the Hodge conjecture for a fibre product of two non-isotrivial families of $\mathrm K3$-surfaces (possibly with degenerations) under the condition that, for every point of the curve, at least one family has non-singular fibre over this point, and the rank of the lattice of transcendental cycles on a generic geometric fibre of one family is odd and not equal to the corresponding rank for the other.

Keywords: Hodge conjecture, standard conjecture of Lefschetz type, $\mathrm K3$-surface.

Funding Agency Grant Number
Russian Foundation for Basic Research 09-01-00132
Dynasty Foundation


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English version:
Izvestiya: Mathematics, 2013, 77:1, 143–162

Bibliographic databases:

UDC: 512.6
MSC: 14C25, 14F25, 14J40
Received: 28.11.2011

Citation: O. V. Nikol'skaya, “On algebraic cycles on a fibre product of families of K3-surfaces”, Izv. RAN. Ser. Mat., 77:1 (2013), 145–164; Izv. Math., 77:1 (2013), 143–162

Citation in format AMSBIB
\by O.~V.~Nikol'skaya
\paper On algebraic cycles on a~fibre product of families of K3-surfaces
\jour Izv. RAN. Ser. Mat.
\yr 2013
\vol 77
\issue 1
\pages 145--164
\jour Izv. Math.
\yr 2013
\vol 77
\issue 1
\pages 143--162

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    This publication is cited in the following articles:
    1. 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
    2. 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
    3. 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
    4. 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
    5. 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
    6. O. V. Oreshkina, “O gipotezakh Khodzha, Teita i Mamforda–Teita dlya rassloennykh proizvedenii semeistv regulyarnykh poverkhnostei s geometricheskim rodom 1”, Model. i analiz inform. sistem, 25:3 (2018), 312–322  mathnet  crossref  elib
    7. 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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