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As is remarked by B. Totaro, R. Thomas essentially proved that the Hodgeconjecture is inductively equivalent to the existence of a hyperplane section,called a generalized Thomas hyperplane section, such that the restriction to itof a given primitive Hodge class does not vanish. We study the relationsbetween the vanishing cycles in the cohomology of a general fiber, and showthat each relation between the vanishing cycles of type (0,0) with unipotentmonodromy around a singular hyperplane section defines a primitive Hodge classsuch that this singular hyperplane section is a generalized Thomas hyperplanesection if and only if the pairing between a given primitive Hodge class andsome of the constructed primitive Hodge classes does not vanish. This is ageneralization of a construction by P. Griffiths.

Generalized Thomas hyperplane sections and relations between vanishing cycles