Generalized Thomas hyperplane sections for primitive Hodge classes

R. Thomas (with a remark of B. Totaro) proved that the Hodge conjecture is essentially equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We show that the relations between the vanishing cycles have the same dimension as the kernel of the cospecialization morphism associated to a curve not contained in the discriminant. Since the restriction of a given primitive Hodge class lies in this kernel, it implies a certain restriction to a generalized Thomas hyperplane section and also to the associated variation of Hodge structure on the complement of the discriminant when we calculate the singularities of the normal function associated to a primitive Hodge class.