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.
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