  Let $X$ be a general complete intersection of a given multi-degree in a
complex projective space. Suppose that the anti-canonical line bundle of $X$ is
ample. Using the cylinder homomorphism associated with the family of complete
intersections contained in $X$, we prove that the vanishing cycles in the
middle homology group of $X$ are represented by topological cycles whose
support is contained in a proper Zariski closed subset $T\subset X$ of certain
codimension. In some cases, we can find such a Zariski closed subset $T$ with
codimension equal to the upper bound obtained from the Hodge structure of the
middle cohomology group of $X$ by means of Gr\"obner bases. Hence a consequence
of the generalized Hodge conjecture is verified in these cases.
