Thomas hyperplane sections for primitive Hodge classes

R. Thomas proved that the Hodge conjecture is essentially equivalent to the existence of a Thomas hyperplane section having only ordinary double points as singularities and such that the restriction of a given primitive Hodge class to it does not vanish. We show that the relations between the vanishing cycles associated to the ordinary double points of a Thomas hyperplane section have the same dimension as the kernel of the cospecialization morphism associated to any curve not contained in the discriminant. Since the restriction of a given primitive Hodge class lives in this kernel, it implies a certain restriction to a 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.