Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases
Tom 65 / 2004
Streszczenie
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 of a smaller multi-degree 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 $T$ of $X$ with certain codimension. In some cases, by means of Gröbner bases, 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 $X$. Hence a consequence of the generalized Hodge conjecture is verified in these cases.
