# Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

Banach Center Publications (2004)

- Volume: 65, Issue: 1, page 227-259
- ISSN: 0137-6934

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topIchiro Shimada. "Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases." Banach Center Publications 65.1 (2004): 227-259. <http://eudml.org/doc/282347>.

@article{IchiroShimada2004,

abstract = {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.},

author = {Ichiro Shimada},

journal = {Banach Center Publications},

keywords = {complete intersection; Fano variety},

language = {eng},

number = {1},

pages = {227-259},

title = {Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases},

url = {http://eudml.org/doc/282347},

volume = {65},

year = {2004},

}

TY - JOUR

AU - Ichiro Shimada

TI - Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

JO - Banach Center Publications

PY - 2004

VL - 65

IS - 1

SP - 227

EP - 259

AB - 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.

LA - eng

KW - complete intersection; Fano variety

UR - http://eudml.org/doc/282347

ER -

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