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1 Banach Center Publications

1 2004

1 Shimada I.

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Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

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Shimada I.

Banach Center Publications

2004

tom 65

nr 1

227-259

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.