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Even-Relative-Dimensional Vanishing Cycles in Bivariant Intersection Theory

Published online by Cambridge University Press:  11 January 2016

Hiroshi Saito
Hiroshi Saito*
Affiliation:
Department of mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya, Japan, saito@math.nagoya-u.ac.jp
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Abstract

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For a smooth variety proper over a curve having a fibre with isolated ordinary quadratic singularities, it is well-known that we have the vanishing cycles associated to the singularities in the étale cohomology of the geometric generic fibre. The base-change by a double cover of the base curve ramified at the image of the singular fibre has singularities corresponding to the singularities in the fibre. In this note, we show that in the even relative-dimensional case, there exist elements of the bivariant Chow group of the base-change with supports in the singularities and hence their images in the bivariant Chow group with supports in the special fibre and that the usual cohomological vanishing cycles are obtained as their images by a natural map, a kind of “cycle map” so that the elements in the bivariant Chow groups can be regarded as the vanishing cycles. The bivariant Chow group with supports in the special fibre has a ring structure and the natural map is a ring homomorphism to the cohomology ring of the geometric generic fibre. Also discussed is the relation of the bivariant Chow group with supports in the special fibre to the specialization map of Chow groups.

Keywords

14C1714F17
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 187 , September 2007 , pp. 49 - 73
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

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