Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T04:33:35.681Z Has data issue: false hasContentIssue false
  • English
  • Français

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, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[B] Bourbaki, N. Éléments de mathématique, Algèbre commutative, Hermann, 1967.Google Scholar
[SGA 7] Deligne, P. and Katz, N., Groupes de monodromie en géométrie algébrique II, Lecture notes in mathematics, vol. 340, Springer-Verlag, Berlin-Heidelberg-New York, 1973.Google Scholar
[F] Fulton, W., Intersection theory, Springer-Verlag, 1984.Google Scholar
[EGA] Grothendieck, A. and Dieudonne, J., Éléments de géométrie algébrique, Publications mathématiques de IHES, No. 4, 8, 11, 17, 20, 24, 28, 32.Google Scholar
[H] Hodge, W. V. D. and Pedoe, D., Methods of algebraic geometry, volume 2, Cambridge University Press, 1952.Google Scholar
[K] Kimura, Shun-ichi, Fractional intersection and bivariant theory, Comm. Algebra, 20 (1992), no. 1, 285–302.Google Scholar