  In this paper we discuss an obstruction to the integral Hodge conjecture,
which arises from certain behavior of vanishing cycles. This allows us to
construct new counter-examples to the integral Hodge conjecture. One typical
such counter-example is the product of a very general hypersurface of odd
dimension and an Enriques surface. Our approach generalizes the degeneration
argument of Benoist-Ottem.
