Geodesic
On algebraic cycles on a~fibre product of families of K3-surfaces
Izvestiya. Mathematics , Tome 77 (2013) no. 1, pp. 143-162

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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, $\mathrm K3$-surface.
Mots-clés : standard conjecture of Lefschetz type 
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     author = {O. V. Nikol'skaya},
     title = {On algebraic cycles on a~fibre product of families of {K3-surfaces}},
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O. V. Nikol'skaya. On algebraic cycles on a~fibre product of families of K3-surfaces. Izvestiya. Mathematics , Tome 77 (2013) no. 1, pp. 143-162. http://geodesic.mathdoc.fr/item/IM2_2013_77_1_a6/