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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc65-0-15,
author = {Ichiro Shimada},
title = {Vanishing cycles, the generalized Hodge Conjecture and Gr\"obner bases},
journal = {Banach Center Publications},
volume = {65},
year = {2004},
pages = {227-259},
zbl = {1058.14015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc65-0-15}
}
Ichiro Shimada. Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases. Banach Center Publications, Tome 65 (2004) pp. 227-259. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc65-0-15/