Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

Ichiro 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 -