On Algebraic Cycles on Fibre Products of Non-isotrivial Families of Regular Surfaces with Geometric Genus 1 - O. V. Nikol’skaya (Моделирование и анализ информационных систем)
Let      ) be a projective family of surfaces (possibly with degenerations) over a smooth projective curve  . Assume that the discriminant loci       are disjoint,          for any smooth fibre     and the period map associated with the variation of Hodge structures         (where             is a smooth part of the morphism    ), is non-constant. If for generic geometric fibres     and     the following conditions hold: (i)         is an odd integer; (ii)              , then for any smooth projective model   of the fibre product         the Hodge conjecture on algebraic cycles is true. If, besides, the morphisms     are smooth,           are odd prime numbers and      , then for