  As is remarked by B. Totaro, R. Thomas essentially proved that the Hodge
conjecture is inductively 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 study the relations
between the vanishing cycles in the cohomology of a general fiber, and show
that each relation between the vanishing cycles of type (0,0) with unipotent
monodromy around a singular hyperplane section defines a primitive Hodge class
such that this singular hyperplane section is a generalized Thomas hyperplane
section if and only if the pairing between a given primitive Hodge class and
some of the constructed primitive Hodge classes does not vanish. This is a
generalization of a construction by P. Griffiths.
